Introduction to Holomorphic Functions of Several Variables, Volume I: Function Theory

Citation preview

INTRODUCTION TO HOLOMORPHIC FUNCTIONS OF SEVERAL VARIABLES Volume I: Function Theory

Robert C. Gunning

WADSWORTH &BROOKS/COLE MATHEMATICS SERIES

Introduction to Holomorphic Functions of Several Variables Volume I Function Theory

Robert C. Gunning Princeton University

w~ Wadsworth & Brooks/Cole Advanced Books & Software

Brooks/Cole Publishing Company A Division of Wadsworth, Inc.

© 1990 by Wadsworth, Inc., Belmont, California 94002. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed, in any form or by any meanselectronic, mechanical, photocopying, recording, or otherwise-without the prior written permission of the publisher, Brooks/Cole Publishing Company, Pacific Grove, California 93950, a division of Wadsworth, Inc. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging in Publication Data Gunning, R. C. (Robert Clifford), [date] Introduction to holomorphic functions of several variables / Robert Gunning. p. cm. Revised version and complete rewriting of: Analytic functions of several complex variables. Includes bibliographical references. Contents: v. 1. Function theory-v. 2. Local theory-v. 3. Homological theory. ISBN 0-534-13308-8 (v. 1).-ISBN 0-534-13309-6 (v. 2).-ISBN 0-534-13310-X (v. 3) 1. Holomorphic functions. 2. Functions of several complex variables. I. Gunning, R. C. (Robert Clifford) Analytic functions of sev~J 1, are automatically removable singularities for holomorphic functions in general, with no hypotheses of boundedness. Several more examples of removable singularities arc given; these can be skipped if desired, but they do provide some useful intuitive feel for a property that is peculiarly characteristic of holomorphic functions of several variables. E. The differential operator that appears in the analogue for functions of several variables of the Cauchy-Riemann characterization of hoi omorphic functions can be extended as a linear mapping from complex differential forms of bidegree (p, q) to those ofbidegree (p, q + 1) and has the property that aa = o. Thus if ¢J = at/!, then a¢J = 0,

a

xii

Volume I. Function Theory

xiii

and that raises the question whether, conversely, if a¢ = 0, then there is a differential form", such that a", = ¢. That is true locally and also for differential forms in such simple domains as polydiscs. What may seem to be a digression here is actually not one; this result has a very nice application to another removable singularities theorem, and this is in turn the model for a deep and powerful general use of the a operator in complex analysis, to be discussed in section o. F. Another application of properties of this differential operator yields an interesting and useful result about the approximation of holomorphic functions in special domains in en by polynomials, in extension of the Runge approximation theorem for functions of a single variable. For several variables, though, there can be such approximation theorems only for somewhat special domains, as indicated by an example. G. The discussion of removable singularities indicates that there are some pairs of open subsets D ~ E ~ en such that every function holomorphic in D extends to a holomorphic function in E. Particularly natural and useful are those subsets D ~ en for which there is actually no nontrivial such extension-that is, for which necessarily E = D. Such sets are called domains of holomorphy and have a number of special properties that will be considered subsequently. Some alternative characterizations and various examples of domains of holomorphy are provided here. D is not a domain ofholomorphy, it is but natural to ask for the maximal set E to which all holomorphic functions in D extend. A complication is that the extended functions may be multiple-valued; but all of them can be viewed as single-valued holomorphic functions on a complex manifold spread out as a locally unbranched covering space over an open subset of en. In the category of such manifolds, called Riemann domains, there is a unique maximal one to which all holomorphic functions in D extend, the envelope of holomorphy of D; it can be described intrinsically in terms of the ring of holomorphic functions on D. I. An envelope of holomorphy is itself maximal, the analogue of a domain of holomorphy among Riemann domains, virtually by definition. It is possible to extend many of the alternative characterizations of domains of holomorphy to hold on Riemann domains as well, as is done here with elementary but in part quite nontrivial arguments. A more complete and in many ways preferable treatment of this topic is given in section P, after some more machinery is developed, so this section can be omitted altogether by those who are willing to proceed further; it is really inserted just to conclude the discussion for those wishing to go no further in this direction. The preceding portion is by itself a reasonable introduction to function theory in several variables. It is quite possible for those so inclined to omit the rest of this volume, and proceed to either Volume II or Volume III. On the other hand, those interested in functions as such may wish to plow on through the rest of Volume I, which concentrates on a particularly useful class of nonholomorphic functions. J. Plurisubharmonic f.lOctions have played a considerable role in function theory of several variables. They are natural and useful extensions to several variables of the possibly familiar subharmonic functions of a single complex variable. A review of the relevant properties of subharmonic functions is included here for those who may not be that well acquainted with them. K. The basic properties of plurisubharmonic functions are discussed in some detail. L. Various special or generalized classes of plurisubharmonic functions are useful at some points and so

".If

xiv

Volume I. Function Theory

are discussed. It is quite possible, and perhaps at first reading advisable, to skip this section until the need arises in references to it later. M. There is a special class of open subsets of en described in terms of the plurisubharmonic functions in them, the pseudoconvex subsets. These are in some ways analogous to domains of holomorphy. Indeed, it is demonstrated in section P that they are precisely domains of holomorphy, although that is quite a nontrivial result. These sets have a number of special properties and alternative characterizations, all rather simple to demonstrate because of the flexibility and abundance of plurisubharmonic functions. If it is demonstrated that pseudoconvex subsets are really just domains of holomorphy, these special properties and alternative characterizations hold immediately for the latter subsets; that will yield some very useful and powerful results about domains ofholomorphy. N. Similar arguments hold for Riemann domains, although with some further complications. The discussion here can be omitted altogether by those readers interested in only subsets of en rather than general Riemann domains. O. The main technical tool needed to establish the equivalence of pseudoconvexity and holomorphic convexity is the solvability of the equation in pseudoconvex domains, which is established here. P. The equivalence and some of the immediate consequences are demonstrated quite easily. Q. The preceding results rest eventually on the close relationship between plurisubharmonic and holomorphic functions; some other facets of this relationship are discussed here. R. In conclusion, and as an introduction to another vast area of great interest, the special case of domains with smooth boundaries is briefly discussed. A very quick and convenient survey of various other and more detailed results about pseudoconvex sets can be found in [19]. Powerful tools in studying domains with smooth boundaries are versions of the Cauchy integral formula that involve integrating over the whole boundary; such formulas and their consequences are nicely treated in [48]. Closely related to this is the finer analysis of the operator in smoothly bounded domains, in particular in connection with boundary value problems; this is treated in [17] and [25], where more references to the literature can also be found. There are analogues of other classical boundary value problems in function theory of a single variable treated, for instance, in [50] and [51]. There is an extensive theory of holomorphic mappings and differential-geometric methods, as discussed, for instance, in [35].

a

a

Volume II. Local Theory

A holomorphic or meromorphic function of a single variable has the appealingly simple form zn in suitable local coordinates, for some integer n. The zero set of a holomorphic function of one variable or the singular set of a merom orphic function of one variable is an isolated point. The situation is much more complicated and hence much more interesting for functions of several variables. It leads to an extensive collection of results in a quite different direction from that of Volume I, indeed more akin to algebraic geometry than to classical function theory. A. A basic tool for the local study of hoi omorphic functions of several variables is a convenient canonical form for those functions, the Weierstrass polynomials. From that it is easy to derive some general properties of the ring oflocal holomorphic functions at a point in en. B. A holomorphic subvariety of an open subset in en is a subset that can be described locally as the set of common zeros of finitely many holomorphic functions. Such subvarieties are described locally in terms of ideals in the ring of local holomorphic functions, with interesting simple relations between the algebraic and geometric structures. It is convenient to consider some natural equivalence classes of holomorphic subvarieties as abstract holomorphic varieties. C. The simplest local form of a holomorphic mapping between holomorphic varieties is a finite branched holomorphic covering; any mapping is of this form in the case of a single variable. The general topological and algebraic properties of such mappings are considered first. D. It is demonstrated that every holomorphic variety can be represented locally as a finite branched holomorphic covering of en, when it is locally irreducible. This provides a very convenient local description of holomorphic varieties. E. This description is used to establish some basic local properties of holomorphic varieties, such as the analytic form of Hilbert's zero theorem. F. There are some v\;ry subtle but important properties of holomorphic functions and varieties that are really semilocal rather than local, in the sense that they involve relations between the local rings of hoi omorphic functions at all points of an open neighborhood of a fixed point. These are perhaps the most complicated results of the local theory, but they playa major role in its development. These results are reformulated as coherence theorems in Volume III; they can really be used without worrying about the proofs, although that approach to mathematics has both dangers and drawbacks and should not be encouraged. xv

xvi

Volume II. local Theory

G. Perhaps the basic invariant attached to a holomorphic variety is its dimension. This can be characterized either geometrically or algebraically. There are a number of relations between the dimension of a subvariety and the number of holomorphic functions needed to describe the subvariety. A useful digression here is the description of general divisors and divisors of functions. The results obtained so far in this volume provide an introduction to the local theory of holomorphic functions and varieties, quite sufficient for many purposes. It is possible for those so interested to skip the remainder of this volume and pass to the global theory in Volume III. On the other hand, though, a great deal more can be said about the local theory. H. It is natural to consider hoI om orphic functions on holomorphic varieties, but it is not altogether trivial to show that a limit of such functions is also holomorphic. That closure property and an extension to the closure of modules of various sorts are useful and important, if somewhat technical, results. I. Another useful local invariant is not the dimension of a variety itself but rather the least value m such that the variety can be realized as a subvariety of Cm ; this is known as the tangential or imbedding dimension of the variety. This invariant is strictly larger than the ordinary dimension at singular points and is indeed a measure of the singularity. It can be described in terms of the natural notion of the tangent space to a variety, even at singular points, and is useful in examining differential notions such as the inverse mapping theorem at singular points. J. Global sections of the tangent bundle, or holomorphic vector fields, can be introduced on varieties with singularities, and they playa role much like the one they play on manifolds. K. With the additional machinery that has been developed here, it is possible to complete the discussion of holomorphic extensions of holomorphic functions from section D of Volume I. There are analogous results for the holomorphic extension of holomorphic varieties. L. Finite holomorphic mappings were discussed in section C. Local properties of general holomorphic mappings are examined in some detail here. The simplest classes of holomorphic mappings are those for which the dimension of the inverse image of a point is independent of the point; they have relatively simple standard representations, extending those examined earlier for finite holomorphic mappings. M. For completeness, and in preparation for the further discussion of local properties of holomorphic mappings, a review of some standard properties of complex projective spaces is given. This digression can be skipped by readers already familiar with such spaces. N. The discussion of holomorphic mappings continues with an analysis of proper holomorphic mappings, those for which the inverse image of a compact set is compact. The images of these mappings, and of those of a natural generalization of this class of mappings, are always holomorphic varieties themselves, a useful and nontrivial result. Among these mappings is the special class of monoidal or quadratic transforms, which playa major role in the detailed analysis of singularities of holomorphic varieties. O. Meromorphic functions were earlier considered locally; here they are examined globally, first in open subsets of C". P. The extension of this discussion to holomorphic functions on varieties is more complicated. Q. There is a special class

Volume II. Local Theory

xvii

of varieties of great importance here; those for which the bounded meromorphic functions are necessarily holomorphic. They are called normal varieties. R. To any holomorphic variety there is naturally associated a unique one-sheeted branched covering that is itself a normal variety. This normalization is a convenient and useful partial step in the classification of singularities of holomorphic varieties. Further discussion of general properties of hoI om orphic varieties with singularities and other references to the literature can be found in [1], [16], [23], [28], [43], and [60]. The topological properties of singularities of one-dimensional varieties are a classical topic in algebraic geometry, while the higher-dimensional study was pioneered in [41]. The case of two-dimensional singularities is the next most extensively studied class, as surveyed, for instance, in [37]. For many purposes it is important to study holomorphic varieties with more general rings of holomorphic functions than those considered here, rings with nilpotent elements, reflecting a description of subvarieties by particular families of defining equations; these topics are surveyed nicely in [23].

Volume III. Homological Theory

Sheaves have proved to be very useful tools in organizing and simplifying various arguments and calculations in function theory, as well as in algebraic geometry, and should be part of the equipment of those working in this area. A. The definitions and basic properties of sheaves are developed in a fairly general form, assuming no previous acquaintance; those readers already familiar with sheaves can skip or skim through this section. B. Sheaves provide a particularly convenient mechanism for handling the semilocal properties of holomorphic functions and varieties discussed in the preceding volume. The notion of coherence plays a prominent role here. C. Sheaves provide an equally convenient mechanism for handling some of the basic global problems of complex analysis through cohomology theory. Again no previous acquaintance with this theory is presupposed. The algebraic structure underlying cohomology theory is developed first, including some general properties of cohomology theories and the technique of spectral sequences. D. The cohomology groups of a space with coefficients in a sheaf are examined in some detail, together with techniques for calculating these groups by using convenient auxiliary sheaves. E. There is another technique for calculating these groups, a useful technique in some analytic applications; it involves sections of the sheaf over the sets forming various open covers of the space. F. The behavior of sheaves and sheaf cohomology groups under mappings of the base spaces is examined only as necessary for applications here. G. The discussion turns next to the use of sheaf cohomology in complex analysis, A basic technical tool is developed first, a result about matrices of holomorphic functions, a variant of the possibly familiar Riemann-Hilbert problem. H. This auxiliary result is used to show that an open polydisc has trivial cohomology groups in all positive dimensions, when the coefficient sheaf is any holomorphic sheaf satisfying the basic semilocal coherence condition. I. The vanishing of all of these cohomology groups has a wide range of useful and interesting consequences. In general, a holomorphic variety such as a polydisc for which these cohomology groups vanish is called a Stein variety. Some of the fundamental properties of Stein varieties, including approximation results reminiscent of the Runge theorem, are established first. J. The special properties of the Frechet algebra of holomorphic functions on a Stein variety are examined. K. The global properties of meromorphic xviii

Volume III. Homological Theory

xix

functions on a Stein variety, and their relationship to holomorphic functions, are discussed. L. Alternative characterizations of Stein varieties are established to show that there actually do exist a wide range of interesting such varieties. For instance, an open subset of en is a Stein variety precisely when it is a domain of holomorphy; any subvariety of a Stein variety is itself a Stein variety, and certain increasing unions of Stein varieties are Stein varieties. The results established up to this point in Volume III provide a good introduction to the use of homological methods in complex analysis, as well as to the properties of Stein varieties, so some readers may be content to stop here. There are, however, further interesting properties and alternative characterizations of Stein varieties. M. For instance, any holomorphic variety that admits a finite proper holomorphic mapping into a Stein variety is itself a Stein variety. That generalizes the earlier observation that any subvariety of a Stein variety is a Stein variety, and can be used to simplify some of the other characterizations of Stein varieties discussed in the preceding section. N. For another instance, a hoi om orphic variety is Stein if and only if its normalization is Stein; more generally, if F: V -+ W is a finite branched holomorphic covering, then V is Stein if and only if W is Stein. O. There are in addition various cohomological characterizations of Stein varieties; some of these have applications to the problem of the number of holomorphic functions needed to describe a holomorphic subvariety. P. One of the basic properties of Stein varieties, as made manifest in the preceding discussion, is that they have a plenitude of global holomorphic functions. Indeed, it is noted here that a general n-tuple of holomorphic functions on an n-dimensional Stein variety V determines a finite holomorphic mapping from V into en. Q. Moreover, a dense set of (n + I)-tuples of hoi om orphic functions on V determines a proper holomorphic mapping from V to en +1 • R. These results can be , combined to yield yet more characterizations of Stein varieties, perhaps the most appealing and satisfactory of all the characterizations. For instance, a holomorphic variety V is Stein precisely when there is a holomorphic homeomorphism from V to a holomorphic subvariety of some space en. A very important topic in the continuation of the discussion of global methods is another solution of the Levi problem using sheaf-theoretic techniques, due to Grauert; this was discussed in the first version of the present book. It is possible to extend the notion of plurisubharmonic functions to general holomorphic varieties as well and approach the Levi problem this way. These topics have not yet appeared very accessibly in textbooks, but a useful survey of the literature can be found in [6]. Other topics are treated in [4] and [24], among other places. There are extension theorems for hoi om orphic sheaves, analogues of the extension theorems for holomorphic functions and varieties, discussed in [54].

A Elementary Properties of Holomorphic Functions

The field of real numbers is denoted by ~ and the field of complex numbers by C. Both are topological fields with well-known properties. In studying the theory of functions of several complex variables, the space of primary interest is the n-dimensional complex vector space en, which can be identified with the Cartesian product C x ... x C ofn copies of the complex plane, or with the ordinary Euclidean space ~2n of dimension 2n. A point of cn is denoted by Z = (z l' ... , zn), where Zj = Xj + iYi' with Xj and Yj real numbers and i a fixed square root of - 1. In considering the topology of en, two special bases for the open subsets are particularly useful. An open polydisc in en is a subset of the form

and is thus the Cartesian product of n open discs. The point A E Cn is called the center of the polydisc, and the point R = (r 1, ... , rn) E ~n is called its polyradius. When r 1 = ... = rn' this common value r is sometimes also called the radius of the polydisc, and the polydisc is denoted by A(A; r). For some purposes is is convenient to consider as an open polydisc in the extended sense one for which some of the radii rj may be 00; otherwise, 0 < rj < 00 for 1 ~ j ~ n. As usual, Izi denotes the modulus or absolute value of the complex number z. An open ball in en is a subset of the form B(A; r) = B(a 1 ,

••• ,

an; r)

The point A E C" is called the center of the ball, and the positive real number r is called its radius. This set is, of course, a ball in the usual sense in terms of the Euclidean metric liZ -_All = (Ljlzj - aj I2 )1/2. The point set closures of these sets are the closed polydisc A(A; R) and the closed ball .8(A; r), defined respectively by

2

Volume I

Function Theory

and

Again for some purposes it is convenient to consider as a closed polydisc or closed ball in the extended sense one for which some rj = 0 or r = 0; otherwise, 0 < rj < 00 and 0 < r < 00. When n = 1, the polydisc and the ball coincide. A complex-valued function f on a subset D ~ en is merely a mapping from D to the complex plane; the value of the function f at a point ZED is denoted by f(Z) as usual. The extension to several variables of the concept of a holomorphic function of one variable is perhaps most naturally achieved as follows, using what is really the basic property of holomorphic functions. 1. DEFINITION. A complex-valued function f defined on an open subset D ~ en is holomorphic in D if each point A E D has an open neighborhood U such that the function f has a power series expansion 00

f(Z)

=.

~

Ci!"'iJZI - al)i! ... (zn - an)i n

(1)

'1,0 .. ,l n =0

which converges for all Z by (!)D'

E

U. The set of all functions holomorphic in D is denoted

The multi-index notation is frequently used to simplify formulas involving power series in several variables. If I = (iI' ... , in) E zn and Z = (z I, ... , zn) E en define Zl = zi!'" z!n, and (1) can be written in the form (1 ') It is a familiar result from elementary analysis that if the power series (1) converges at some point BEen, then it converges absolutely and uniformly in any open polydisc L1(A; R) for which rj < Ibj - aJ The standard proofs of this assertion for power series

in one variable extend readily to the case of power series in several variables. A first consequence of this observation is that the function f is continuous in any such polydisc, as the uniform limit of continuous functions; hence, any function holomorphic in D is necessarily continuous in D. A second consequence is that the power series (1) can be rearranged arbitrarily; thus, if the coordinates Z I' ... , Zj-l' Zj+l, ... , Zn are given some fixed values b l , ... , bj- l , bj + l , ... , bn> then the power series (1) can be rearranged as a convergent power series in Zj - aj. That is to say, a function f

E (!)D

is holomorphic in each variable separately throughout the domain D

in the sense that f(b l , ... , bj - l , Zj' bj+ l , ... , bn) is holomorphic as a function of Zj for any fixed values bi whenever (b l , ... , bj - l , Zj' bj+ l , ... , bn ) E D. The converse is also true but not at all trivial and will be proved in the next section; however, a partial converse at least is easily demonstrated.

A

Elementary Properties of Holomorphic Functions

3

2. LEMMA (Osgood's lemma). If a complex-valued function is continuous in an open subset D s;: en and is holomorphic in each variable separately, then it is holomorphic in D. Proof. Choose any point A E D and closed polydisc L\(A; R) s;: D. Since f is holomorphic in each variable separately in an open neighborhood of L\(A; R), repeated application of the Cauchy integral formula for holomorphic functions of one variable leads to the formula

for all Z E L\(A; R). For any fixed point Z the integrand in (2) is continuous on the compact domain of integration, so the iterated integral in (2) can be replaced by the single multiple integral (3)

Now for any fixed point Z

E

L\(A; R), the series expansion

is absolutely and uniformly convergent for all points' on the domain of integration; so upon substituting this series expansion into (3) and interchanging the orders of summation and integration, it follows that f(Z) has a convergent series expansion of the form (1) in L\(A; R), where

(4) That concludes the proof. Some ofthe observations made during the course of the preceding proof merit separating out for special attention. First, any function f that is holomorphic in an open neighborhood of a closed polydisc L\(A; R) has a Cauchy integral formula of the form (3) inside that polydisc. This is the simplest and most direct extension of the Cauchy integral formula for holomorphic functions of one variable, but it differs substantively from the formula for the case offunctions of one variable; for example, if n > 1, the integration in (3) is not extended over the full boundary of the polydisc L\(A; R), a set of topological dimension 2n - 1, but only over an n-dimensional part of that boundary. Next, since a holomorphic function of several variables is holomorphic in each variable separately, it is possible to apply the usual complex derivative to each variable Zj separately; the resulting differential operator is denoted by ojOZj' From these differential operators applied to (3), the Cauchy integral representation formulas for the derivatives follow in the form

4

Volume I

Function Theory

(5)

Upon comparing equations (4) and (5), it follows that (6)

The same observation results from term-by-term differentiation of the series (1). These are also formulas that can be simplified by use of the multi-index notation. If I = (iI' ... , in), then set III = i l + ... + in and I! = (il!)···(in !). The preceding equations can be rewritten (5')

and (6') Finally, note as a consequence of (6) that the power series expansion (1) is uniquely determined by the restriction of the function f to any open neighborhood of the point A. If f is holomorphic in a domain D ~ this power series expansion necessarily converges in any polydisc L\(A; R) ~ D by the construction used in the proof of Lemma 2; indeed, this series converges absolutely and uniformly in any closed polydisc

en,

A(A; R) ~ D. It is convenient to introduce here the linear first-order partial differential

operators

o

1(0 .0) oYj

---

--1-

OZj - 2 oXj

and

o

1(0-+ i -0)

-=-

OZj

2 oXj

oYj

(7)

As is well known, the classical Cauchy-Riemann criterion that a continuously differentiable complex-valued function f(x j , Yj) of two real variables be a holomorphic function of Zj = Xj + iYj is that of/ozj = 0 throughout the region of definition of the function; for a hoi om orphic function f(zj), the expression of/ozj coincides with the complex derivative of the function f These assertions extend immediately to functions of several variables. Thus by Osgood's lemma, a function f(x l ' Y I ' ... , Xn, Yn) that is continuously differentiable in a domain D ~ en is a holomorphic function in D precisely when Of/OZl = ... = of/ozn = 0 throughout D. It is customary to refer to these equations as the Cauchy-Riemann equations for several variables. Several other results also extend immediately from one to several variables by Lemma 2. Although some of these extensions are quite trivial, it is worth pointing

A Elementary Properties of Holomorphic Functions

5

them out explicitly here so that they can be used later without further comment. Thus, if D is an open subset of en, then (!)D is a ring under pointwise addition and multiplication of functions; indeed, it is an algebra over the complex numbers. This amounts merely to the assertion that the sum or product of two holomorphic functions is holomorphic.If f E (!)D and f is nowhere zero, then 1/f E (!)D; the nowhere vanishing functions thus form a group under multiplication, and this group is denoted by (!)Z- If f E (!)D and f is real-valued or has constant modulus, then f is locally constant, for f is locally constant in each variable separately. If a sequence of functions f E (!)D converges uniformly on compact subsets of D, then the limit function is also holomorphic on D, for the limit function is continuous and is hoI om orphic in each variable separately. Finally, hoI om orphic functions are closed under composition, although the meaning of that statement in the case of functions of several variables requires a few more words of explanation. A mapping F: D -+ D' from an open domain D s;; en into an open domain D' s;; en' can be described by its n' coordinate functions Wj = jj(Zl' ... , zn) for 1 ~j ~ n', and it is called a holomorphic mapping if these coordinate functions are hoI om orphic in D. If g E (!)D' and F: D -+ D' is a holomorphic mapping, then the composition go F(Z) = g(F(Z)) is a holomorphic function in D; indeed, the mapping g -+ g 0 F is a ring homomorphism, even an algebra homomorphism,

To verify that g 0 F is holomorphic, note first that when g and F are merely continuously differentiable functions of the real variables uj ' Vj and Xj' yj' respectively, where Wj = uj + iVj and Zj = Xj + iYj' then from the usual chain rule for differentiation it follows immediately that

This is the complex form of the chain rule for differentiation. If F is hoI om orphic, then OWk/~ = 0, and if g is holomorphic, then og/owk = O. If both are holomorphic, then o(g 0 F)/~ = 0 and g 0 F is holomorphic as desired. Some other results have fairly easy extensions from one to several variables, but the extensions require slightly more justification.

3. THEOREM (identity theorem).

If f and g are holomorphic functions in a connected open subset D s;; en and if f(Z) = g(Z) for all points Z in a nonempty open subset U s;; D, then f(Z) = g(Z) for all points Z in D.

6

Volume I Function Theory

Proof. Let E ~ D be the interior of the set consisting of all points Z for which f(Z) = g(Z); thus, E is an open subset of D and is nonempty, since U ~ E. Because D is connected, in order to complete the proof of the theorem it suffices merely to show that E is relatively closed in D. If A E D n E, where E denotes the point set closure of E, then choose a positive number r sufficiently small that A(A; r) ~ D and choose a point B E ~(A; r/2) n E, noting that there must exist such a point since A E E; then A E ~(B; r/2) ~ D. The function f - g has a power series expansion centered at B and converging at all points in the polydisc ~(B; r/2); but since f - g is identically zero in an open neighborhood of B, the coefficients in that power series expansion are all zero by (6). Hence, f(Z) - g(Z) = 0 for all points Z E ~(B; r/2), and consequently A E ~(B; r/2) ~ E. That suffices to conclude the proof of the theorem. 4. THEOREM (maximum modulus theorem). If f is holomorphic in a connected open subset D ~ en and if there is a point A ED such that If(Z)1 ~ If(A)1 for all points Z in some open neighborhood of A, then f(Z) = f(A) for all points ZED.

Proof. In the pattern of one of the customary proofs of the maximum modulus theorem for functions of one variable, note as a consequence of the Cauchy integral formula (3) that for any open polydisc ~ = ~(A; R) ~ D for which ~ ~ D,

1~lf(A) =

L

f(Z) dV(Z)

where dV(Z) is the Euclidean volume element in is the Euclidean volume of ~; therefore,

nnrt· .. r;

1~1'lf(A)1 ~

en =

1R 2 n and I~I = J r, but the question of its convergence on the boundary circle Izl = r is generally a rather complicated one. That suggests that when examining the set of points at which a power series in several complex variables converges, it is convenient to consider initially the largest open set in which the series converges, the domain of convergence of the power series. A point Z E en is in the domain of convergence precisely when the power series converges at all points in an open neighborhood of Z. A power series necessarily converges absolutely at each point of its domain of convergence. A convenient terminology used in the discussion of such domains of convergence is the following: 1. DEFINITION. An open subset D ~ en is a Reinhardt domain if, whenever (z l' ... , zn) E D then «(1 z 1, ... , (nzn) E D for all complex numbers (j with !(jl = 1. It is a complete Reinhardt domain if, whenever (z 1, ... , zn) E D, then «(1 z 1, ... , (nZn) E D for all complex numbers (j with !(jl ~ 1. The base of a Reinhardt domain D is the set {(I Z11, ... , IZnl): (ZI' ... , zn) ED}.

A Reinhardt domain is sometimes also called a circled domain, since it is obviously a union of circles. The complete Reinhardt domains are just those open subsets of en that can be written as unions of open polydiscs centered at the origin. The base of a Reinhardt domain is a relatively open subset of the closed positive cone {(r 1 , ••• , rn) E ~n : rj ~ O}, but it is not necessarily an open subset of ~n, since it may contain some boundary points (rl' ... , rn) with rj = O. The base completely determines the Reinhardt domain, since if B is the base of a Reinhardt domain D, then

B Convergence Properties of Power Series

11

----,R R' I • I

_+-__IL...-_ _

---'L...--;~

Izd

--t-----'L....-...l.....--_ _ _

Izd

Figure 1

This is a particularly simple description of the domain D in the case n = 2 because then the base is a relatively open subset of the closed positive quadrant of the Euclidean plane, as in Figure 1 for example, and is easily visualized. A Reinhardt domain with base B is complete precisely when B contains together with any point R all the points R' such that 0 ~ rj ~ rj for all j. The Reinhardt domain with base BI as sketched in Figure 1 is complete, whereas that with base B2 is not, for example. For some purposes it is also convenient to consider the set

Note that this is a subset of the full space ~n rather than just of the closed positive cone, and that log B is naturally homeomorphic to the interior of the base B. The ~et B is called logarithmically convex if log B is a convex subset of ~n in the usual sense.

2. THEOREM.

The domain of convergence of a power series in n complex variables is a complete Reinhardt domain having a logarithmically convex base.

Proof. If a power series L C1Z1 converges at a point A E en, then it converges at all points of the open polydisc ,1(0; lall, ... , lanl), so the domain of convergence is obviously a complete Reinhardt domain. If B is the base of this Reinhardt domain, then to show that the open set log B is convex, it suffices merely to show that whenever Rand S belong to B and have no zero components, then T also belongs to B, where log tj = !(log rj + log Sj) or, equivalently, where tj = r1'2sjl2. If R, S E B, then the series LcIRI and LCISI converge, but since

by the inequality of the arithmetic and geometric means, the series converges, and consequently T E B. That concludes the proof.

L CI Tl also

A simple corollary of this theorem exhibits one of the most interesting differences between holomorphic functions of one and of several variables. If D is

12

Volume I

Function Theory

a complete Reinhardt domain with a base B that is not necessarily logarithmically convex, introduce the convex hull A of the set log B = A, the smallest convex set containing A, and let

so that

Bo is an open subset of [R" for which A =

log

Bo. The set

is the base of a complete Reinhardt domain fj s;; C", which is called the Reinhardt hull of D. Note that Bo contains the interior of the base B; hence, B;;;2 Band fj ;;;2 D. Clearly, the set fj may be properly larger than D.

3. COROLLARY.

If D is a complete Reinhardt domain in C", then any function holomorphic in D extends to a holomorphic function in the Reinhardt hull fj of D.

Proof. Any function holomorphic in D has a power series expansion about the origin, and that series converges at all points of D, since D is a complete Reinhardt domain and hence a union of open polydiscs centered at the origin. It follows immediately from the proof of the preceding theorem that this series converges at all points of the Reinhardt domain with base Bo and therefore must also converge at all points of the complete Reinhardt domain fj with base B. That suffices to conclude the proof.

When n = 1, it is evident that fj = D, so the preceding corollary is quite trivial in that case. However, when n > 1, it is evident that there are complete Reinhardt domains D for which the Reinhardt hull fj is properly larger than D. Hence, when n > 1, there are open domains D s;; en such that every function holomorphic in D extends to a holomorphic function in a fixed properly larger domain fj. It is well known, and will be proved again in section G, that this phenomenon cannot occur when n = 1. Indeed, whenever D is an open subset of C1, there exist functions holomorphic in D and not extendable across any boundary point of D. It will also be demonstrated in section G that for any complete Reinhardt domain D with a logarithmically convex base, there exist functions that are holomorphic in D and that cannot be extended across any boundary point of D. That completes the statement of Theorem 2 by showing that any complete Reinhardt domain with a logarithmically convex base is the exact domain of convergence of some power series in n complex variables. It also completes the statement of Corollary 3 by showing that not all functions holomorphic in D can be extended beyond the Reinhardt hull fj of D. It should be noted here that the points of a Reinhardt domain D that lie on the coordinate hyperplanes, the points that have some coordinates zero, play no role in logarithmic convexity; that reflects the fact that they play a somewhat exceptional role in the convergence of power series. For example, if fl and 12 are holomorphic functions in the unit disc A(O; 1) s;; (;1 and do not extend to

B Convergence Properties of Power Series

13

holomorphic functions in any properly larger region, then the domain of convergence of the power series expansion at the origin of the function of two complex variables f(Zl' Z2) = Zd1(zd + Zt!2(Z2) is precisely the polydisc i\(0, 0; 1, 1). But this power series also converges at all points of the spines {(Zl' Z2): Zl = 0, Z2 arbitrary} and {(Zl' Z2): Z2 = 0, Zl arbitrary}. A useful convergence result of a different sort, one that plays a major role in extending Osgood's lemma, Lemma A2, is the following. To simplify the notation, for any point Z = (Zl' ... , z.) E en, let Z' = (Zl' ... , Z.-l) E C·- 1 so that Z = (Z', zn), and correspondingly for any polydisc i\(A; R) s:; C·, write i\(A; R) = i\(A'; R') x i\(a n; rn), where i\(A'; R') s:; en- 1 • 4. LEMMA (Hartogs's lemma). Let f be a holomorphic function in an open polydisc i\(0; R) = i\(O'; R') x i\(0; rn) s:; en, and write the power series expansion of f in this polydisc in the form f(Z)

= L f.(Z')z:

(1)

where f. are holomorphic functions in i\(O'; R'). If there is a constant Yn > rn such that for each fixed point Z' E i\(0; R') the series (1) converges in i\(0; Yn), then the series (1) converges uniformly in any compact subset of i\(0; R') x i\(0; Yn ) and thus extends f to a holomorphic function in this larger polydisc. Proof. Since the functions f.(Z')z: are holomorphic in i\(O'; R) x i\(0; Yn), the last assertion of the lemma is an immediate consequence of the assertion of uniform convergence. To demonstrate the uniform convergence, choose any point A' E i\(O'; R') and any open polydisc i\(A'; S') such that A(A'; S') s:; i\(0; R'), and choose any - constants Sn, sn such that 0 < Sn < rn < sn < Yn • If M ~ 1 is an upper bound for the modulus of the hoi om orphic function f on the compact subset A(A'; S') x .1(0; sn) s:; i\(0; R), then by Cauchy's inequalities, If.(Z') Is: ~ M for all points Z' E A(A'; S') and all indices v. Consequently, for some constant Mo.

1 -loglf.(Z')1 v

~

1 -log M - log v

Sn ~

log M - log

Sn ~

Mo

(2)

for all points Z' E A(A'; S') and all indices v. On the other hand, for any fixed point Z' E A(A'; S'), the series (1) converges in i\(0; Yn), so lim. If.(Z')1 u: = ofor any chosen value Un in the interval sn < Un < Yn • Consequently,

.

1I

lIm .sup ; og If.(Z')1 ~ . -log Un

(3)

for all points Z' E L1(A'; S'). For a slightly smaller polydisc L1(A'; T) c A(A'; S'), the functions (lM 10glf.(Z')1 are uniformly bounded in A(A'; T) by (2). Since these functions are also measurable with respect to the standard Lebesgue measure dV(Z') on cn-1 = 1R 2n - 2 , it follows from Fatou's lemma in measure theory and inequality

14

Volume I

Function Theory

(3) that lim sup •

f

! log If.(Z')1 dV(Z')

~

A(A';T') V

r

lim sup! log II.(Z')I dV(Z')

JA(A';T')

~ -IL\(A';

V

•

T')llog Un

where IL\(A'; T')I = JA(A'; T') dV(Z') is the Euclidean volume of the polydisc L\(A'; T'). For a constant tn in the interval 8n < tn < Un' it follows that there is an index Vo such that

f

! log II.(Z')I dV(Z')

A(A';T') V

~

(4)

-IL\(A'; T')llog tn

whenever v ~ Vo. Since A(A'; T') c L\(A'; S'), if e > 0 is sufficiently small, then for any point W' E L\(A'; e) it follows that L\(A'; T')

S;;;

L\(W'; T'

+ e) S;;; L\(A'; S')

From (2) and (4) note that

f

! 10glf.(Z')1 dV(Z')

A(W';T'+e)

=

v

f

! 10glf.(Z')1 dV(Z')

A(A';T')u[A(W';T'+e)-A(A';T')) V

~ -IL\(A'; T')llog tn + Mo

f

dV(Z')

A(W'; T'+e)-A(A'; T')

Since 8n < tn, it is clear that if e is small enough, the last line in the preceding inequality is at most equal to -IL\(W'; T' + e)llog 8n • Thus,

f

A(W';T'+e)

! 10glf.(Z')1 dV(Z') v

~

-IL\(W'; T'

+ e)llog 8n

Finally, by Jensen's inequality observe that (1M log II.(W')I ~ -log 8n , and hence E L\(A'; e). That in turn implies that the series (1) is absolutely uniformly convergent in L\(A'; e) x L\(O; 8n ). Since A' E L\(O; R') and 8n < in are arbitrary, the series (1) is uniformly convergent on any compact subset of L\(O; R') x L\(O; in). The proof is therewith concluded. If.(W')1 8; ~ 1 for all points W'

Before applying this result to obtain the desired extension of Osgood's lemma, note the following partial extension, which is easily established.

5. LEMMA. If a complex-valued function is uniformly bounded in an open domain D s;;; and is holomorphic in each variable separately, then it is holomorphic in D.

en

Proof. If f is uniformly bounded and holomorphic in each variable separately in D, then it follows immediately from Cauchy's inequalities in each variable separately

B Convergence Properties of Power Series

15

that the complex partial derivatives iJf/iJzj are uniformly bounded on any fixed compact subset KeD. That combined with the Cauchy-Riemann equations implies that the partial derivatives iJfliJxj and iJfliJYj with respect to the underlying real coordinates are also uniformly bounded in K. As is well known, that in turn implies that f is uniformly continuous in K. The desired result then follows immediately from Osgood's lemma, and that concludes the proof. 6. THEOREM (Hartogs's theorem).

If a complex-valued function is holomorphic in each variable separately in an open domain D s en, then it is holomorphic in D.

Proof. The proof will be by induction on the complex dimension n. The theorem is trivial in one dimension, so assume it is true for n - 1 dimensions and consider a function f that is holomorphic in each variable separately in an open domain D s e. It is enough to show that f is holomorphic at any point A E D. Choose an open polydisc A(A; R) centered at A and such that A(A; R) s D. As before, write Z = (Z', zn) for any point Z E e and correspondingly write A(A; R) = A(A'; R') x A(a.; r.).

As a preliminary observation, note that there exist a point b. E A(an; rnl2) and a positive number ~ such that f is uniformly bounded on the polydisc A(A'; R') x A(bn ; ~) s A(A; R). To see that, for any positive integer v, let Xv = {zn

E

A(an; rnI2): If(Z', zn)1 ~ v for all Z'

E

A(A'; R')}

For any fixed Z', the function f(Z', zn) is holomorphic and hence continuous in Zn; that clearly implies that the sets Xv are closed. On the other hand, for any fixed Zn the function f(Z', zn) is holomorphic and hence continuous in Z' by the induction hypothesis, so that If(Z', zn)1 is bounded as Z' varies over the compact polydisc A(A'; R'); that clearly implies that A(an; rnl2) = Uv Xv. It then follows from the Baire category theorem that for some v, the set Xv must contain an open neighborhood A(bn; ~) of some point bn E A(a n; rnI2). Now from Lemma 5 and the preceding observation it follows that f is holomorphic in the polydisc A(A'; R') x A(bn; ~). Choose a constant Sn > rnl2 such that A(b.; sn) s A(an; rn). For each fixed Z' E A(A'; R'), the function f(Z'; zn) is holomorphic in Zn in the disc A(bn; sn). It then follows from Hartogs's lemma (Lemma 4) that f is actually holomorphic in A(A'; R') x A(bn; sn)' and since A E A(A'; R') x A(bn ; sn), the function f is therefore holomorphic near A. That suffices to conclude the proof. It should be observed that Hartogs's theorem is really a result in complex function theory and fails to hold for real analytic functions. Indeed, the traditional example

in ~2 is real analytic except at the origin and is real analytic in each variable

16

Volume I

Function Theory

separately even at the origin, but it is not even continuous at the origin. It should also be observed that from Hartogs's theorem it is possible to characterize holomorphic functions of several variables in ways that are much more analogous to the traditional characterizations of holomorphic functions of one variable. For example, a function f in a domain D £; en is holomorphic in D if the complex partial derivatives

exist at all points of D; or a function f in a domain D £; en is holomorphic in D if it is continuously differentiable with respect to the underlying real coordinates in en = 1R 2n and satisfies the Cauchy-Riemann equations iJf/Ozj = 0 at each point of D.

C Holomorphic Mappings and Complex Manifolds

The inverse and implicit function theorems so familiar from elementary calculus can easily be extended to the corresponding results for holomorphic functions and holomorphic mappings. The extensions can be obtained from the differentiable versions of these theorems merely by showing that the functions of the conclusion are holomorphic if the functions of the hypothesis are, but it is instructive to derive these results by using function-theoretic methods from the outset. If f is a holomorphic function in an open neighborhood of a point A in en and is not identically zero near A, the total order of f at the point A is the least integer v such that some partial derivative of f of order v is nonzero at A. The function f is regular in Zn at the point A if f(a l , .•. , an-I' Zn) is not identically zero as a function of Zn in a neighborhood of an' If f is regular in Zn, then the order of fin z. at the point 'A is the least integer v such that oy/oz: is nonzero at A.

1. DEFINITION.

When the power series expansion of the function f at the point A is written in the form f(Z) =

L f,,(Z) "

(1)

where f" is a homogeneous polynomial of order JI. in the variables Z I - a I' .•• , Zn - an, then the total order of f at the point A is the least integer v such that fv is not identically zero; it is a well-defined nonnegative integer if f is not identically zero near the point A, and it is zero precisely when f(A) # O. The order of fin Zn at the point A is defined only when f(a l , ••• , an-I' Zn) is not identically zero in Zn at the point an, and it is indeed just the order of the zero of the holomorphic function f(a l , ... , an-I' Zn) of the variable Zn at the point an' The order of fin Zn is at least equal to the total order of f but, of course, may be greater.

2. LEMMA.

If f is a holomorphic function of total order v at the origin, then by a suitable nonsingular linear change of coordinates in en that function can be made regular and of order v in zn at the origin. 17

18

Volume I

Function Theory

Proof. The function f has a power series expansion at the origin of the form (1), where fll are homogeneous polynomials of order J.l in the variables Zl ' ... , Zn and v is the least integer such that fv is not identically zero. Choose a point B such that fv(B) #- 0 and a nonsingular linear change of coordinates taking B to the point (0, ... , 0, 1); then in terms of these new coordinates there is a corresponding expansion (1), in which f,.. is identically zero whenever J.l < v but fv(O, ... ,0, 1) #- O. Thus, f is regular and of order v in Zn in terms of these new coordinates, and the proof is thereby concluded. It should be noted incidentally that by the same argument any finite collection of holomorphic functions can simultaneously be made regular of the appropriate orders by a suitable nonsingular linear change of coordinates in en. It is convenient once again to use the notation Z = (Z', zn) for any point of en, where Z' = (z l' ... , Zn-l) E en - 1 , and correspondingly to write any polydisc ~(A; R) in en as the product ~(A; R) = ~(A'; R') x ~(an; rn), where ~(A'; R') c: en - 1 and ~(an; rn) ~ e 1 •

If f is a holomorphic function in a domain D ~ en and is regular and of order v in zn at some point A ED, then there is an open polydisc ~(A; R) = ~(A'; R') x ~(an; rn) with L\(A; R) ~ D such that for any fixed point Z' E ~(A'; R') the function f(Z', zn) as a function of the variable Zn alone has precisely v zeros (counting multiplicities) in the disc A(an; rn) and has no zeros on the boundary of that disc.

3. LEMMA.

Proof. By hypothesis f is holomorphic in some polydisc A(A; S) ~ D and f(A'; zn) as a function of the variable Zn alone has a zero of order v at the point an' Choose a constant rn in the interval 0 < rn < Sn such that f(A', rn) has no zeros in the closed disc L\(a n; rn) except at the point an itself, and let

e=

inf

If(A', zn)1 > 0

(2)

{zn! IZn-anl=rnl

Since f is continuous in an open neighborhood of the compact set {Z = (Z', zn) E en: Z' = A', IZn - ani = rn}, there are constants rj in the intervals 0 < rj < Sj such that If(Z', zn) - f(A', zn)1 < e

whenever

Z' E

~(A'; R'),

IZn - ani

=

rn

(3)

By Rouche's theorem it follows from (2) and (3) that the functions f(A', zn) and f(Z', zn) as functions of Zn alone have the same number of zeros in the disc ~(an; rn); hence, f(Z', zn) has v zeros in that disc as desired. Iff is a holomorphic function in a domain D ~ en and is regular and of order one in zn at some point A E D, then for some open polydisc ~(A; R) = L\(A'; R') x L\(an ; rn) with L\(A; R) ~ D there exists a holomorphic junction gin L\(A'; R') such that

4. THEOREM (implicit function theorem).

i. g(A') = an ii. g(Z') E L\(an; rn) for all Z' E L\(A'; R') iii. f(Z) = 0 for some point Z E L\(A; R) precisely when Zn = g(Z')

C Holomorphic Mappings and Complex Manifolds

19

Proof. Choose an open polydisc A(A; R) ~ D for which the conclusion of Lemma 3 holds. Thus, for any point Z' E A(A'; R') there exists a unique point g(Z') E A(an; rn) such that f(Z', g(Z'» = O. To complete the proof of the theorem it is only necessary to show that the function g just defined is holomorphic in A(A'; R'). For any fixed point Z' E A(A'; R') the function f(Z', zn) has a simple zero at the point Zn = g(Z') and is otherwise nonzero in A(an ; rn ); hence, (Z') g

= _1 2ni

r

en

JI~n-anl~rn f(Z',

en)

af(Z', en) de aen n'

(4)

The function f(Z', en) is nonzero when I'n - ani = rn and Z' E A(A'; R'), so the integrand in (4) is hoi om orphic in Z' in the polydisc A(A'; R'). Therefore, g is also hoi om orphic in that polydisc, thus concluding the proof. By Definition 1, the hypotheses in the implicit function theorem, other than that the function is holomorphic, can be restated in the more traditional form f(A) = 0, (aflazn)(A) #- 0; but the form in which the hypotheses were actually stated suggests a generalization of the theorem, which will be discussed at some length in the next section. The conclusion can in turn be restated as the assertion that the set of zeros of the function f in the polydisc A(A; R) can be described parametrically by Zn = g(Z'); the function g is obviously uniquely determined. In order to extend this result from functions to mappings it is convenient to introduce some additional terminology. If F: D -.. em is a hoi om orphic mapping from an open subset D ~ en into em and is described by the coordinate functions F = (fl' ... ,fm), then the Jacobian matrix of the mapping F at a point A E D is the m x n matrix

The mapping F is nonsingular at the point A ifits Jacobian matrix has maximal rank at that point-that is, if rank JF(A) = min(m, n); the mapping F is nonsingular in D if it is nonsingular at each point of D. When considering a product decomposition en = en- mx em in case n ~ m, a point Z E en will be written Z = (Z', Z") where Z' = (ZI' ... , zn-m) and Z" = (Zn-m+l' ... , zn), and correspondingly the Jacobian matrix at a point A E D ~ en of a holomorphic mapping F: D -.. em will be written JF(A) = (JHA), J;(A» where J;(A) is an m x (n - m) matrix and J;(A) is an m x m matrix. 5. THEOREM (implicit mapping theorem). If F is a holomorphic mapping from an open neighborhood of a point A E en into em for some m ~ n, if F(A) = 0, and if rank J;(A) = m, then for some open polydisc A(A; R) = A(A'; R') x A(A"; R") ~ e.. - mx em = e" there exists a holomorphic mapping G: A(A'; R') -.. A(A"; R") such that G(A') = A" and F(Z) = 0 for some point Z = (Z', Z") E A(A', R) precisely when Z" = G(Z').

Proof. The proof will be by induction on the dimension m. When m = 1, the desired result is exactly that given by Theorem 4, the implicit function theorem; so consider

20

Volume I

Function Theory

some dimension m > 1 and assume that the result has been demonstrated for dimension m - 1. The conclusion involves only the zero locus of the mapping F and so is unchanged when F is replaced by the composition T 0 F of F with a nonsingular linear mapping in em; hence, it can be assumed further that J;(A) = I, the m x m identity matrix. Since iJfm/iJzn = 1 at the point A, it follows from Theorem 4 that for some open polydisc L\(A; R) throughout which the mapping F is holomorphic, there exists a holomorphic mapping

such that g'(a 1 , ••• , an-I) = an and J,,(Z) = 0 for some point Z when Zn = g' (z l' ••• , Zn-1)' The functions

E

L\(A; R) precisely

for 1 ~ i ~ m - 1 are the components of a holomorphic mapping

such that F'(a 1, ... , an- 1) = 0 and

for 1 ~ i, j ~ m - 1. It then follows from the induction hypothesis that, after shrinking the polydisc L\(a 1, ... , an- 1; r1, ... , rn- 1) further if necessary, there exists a hoi om orphic mapping G': L\(A'; R')

~

L\(an- m+1 ,

••• ,

an- 1; rn- m+1,

•.. ,

rn-d

such that G'(A') = (a n- m+1, ••• , an-d and that F'(Zl"'" Zn-1) = 0 for some point (zl"",zn-d E L\(al,· .. ,an-l;r1, ... ,rn-l) precisely when (zn-m+l,,,,,zn-d = G'(Z'). Since F(Z) = 0 for some point Z E L\(A; R) precisely when Zn = g'(Z1,"" zn-d and F' (z l ' ... , Zn-1) = 0, it is clear that the mapping G given by G(Z') = (G'(Z'), g'(Z', G'(Z')))

has the desired properties. That suffices to conclude the proof. 6. THEOREM (inverse mapping theorem). If F is a nonsingular holomorphic mapping from an open neighborhood of a point A E en into en with F(A) = B, then F has a nonsingular holomorphic inverse near B.

Proof. Introduce the holomorphic mapping H from an open neighborhood of the point (B, A) Een x en = e 2n into en defined by H(Z', Z") = F(Z") - Z' whenever Z = (Z', Z") Een x en = e 2n is near (B, A). Note that H(B, A) = 0 and that the

C Holomorphic Mappings and Complex Manifolds

21

matrix { -8h· 8 '. (B, A) : 1 ~ i, j ~ n} zn+J

is nonsingular. It then follows from Theorem 5 that there exists a holomorphic mapping G from an open neighborhood of B into en such that G(B) = A and that H(Z', Z") = 0 for (Z', Z") near (B, A) precisely when Z" = G(Z'); thus, G is a hoI om orphic inverse to F near B. Since FoG is the identity, it follows from the chain rule that I = JFoG(B) = JF(A)· JG(B); hence, JG(B) is invertible so G is nonsingular. That completes the proof of the theorem. A biholomorphic mapping F from an open subset U !:; en onto an open subset en is a holomorphic mapping F: U -+ V which admits a holomorphic inverse.

7. DEFINITION. V

!:;

It should be pointed out here that a biholomorphic mapping is not defined merely to be a one-to-one holomorphic mapping. It is a no doubt familiar observation that real differentiable mappings can be one-to-one without admitting differentiable inverses, as for example is the mapping f: ~l -+ ~l defined by f(x) = x 3 ; its inverse is not differentiable at the origin. It is actually the case that one-to-one holomorphic mappings between open subsets of en necessarily have holomorphic inverses anci hence are really biholomorphic mappings; that result is not altogether obvious, though, and will be proved in Volume II. For the present, biholomorphic mappings are defined to have holomorphic inverses. If F: U -+ V is a biholomorphic mapping with holomorphic inverse G: V -+ U, then the composition G 0 F : U -+ U is the identity mapping; so from the chain rule for differentiation, as in the proof of the preceding theorem, it follows that F and G are both nonsingular throughout their respective domains. Conversely, if F: U -+ V is a nonsingular hoI om orphic mapping, then each point A E U has an open neighborhood UA such that the restriction FI UA : UA -+ F(UA ) is a biholomorphic mapping from UA to the open neighborhood F(UA ) of F(A).

en

If F is a holomorphic mapping from an open domain D !:; into em and if the rank of the Jacobian matrix J F is a constant k throughout D, then for each point A ED there exist open neighborhoods UA of A and UB of B = F(A) and biholomorphic mappings GA : UA -+ ~ and GB : UB -+ VB taking the points A and B to the origin and such that the composite mapping GB 0 FoGAl: VA -+ VB has the form

8. THEOREM.

Proof. To simplify the notation suppose that A and B are both the origin. Then after nonsingular linear changes of coordinates in the domain and range of F, it can further be assumed that

rank{~ (A): 1 ~ i,j ~ k} = k

22

Volume I Function Theory

Now set f; = fl' ... , fk = h and choose other functions fk+l' ... , f: holomorphic in an open neighborhood of the origin and such that (A)'. 1 < . .< rank { of;' OZj = I,J = n} - n

These functions can be taken as the coordinate functions of a nonsingular holomorphic mapping GA in an open neighborhood of A, and for this mapping it is evident that the composite FoGAl has the form

Moreover, since the rank of the Jacobian matrix of the mapping FoGAl is also identically equal to k, it is apparent that the functions fk~l' ... , f:: depend only on the variables Z 1, ... ,Zk' Finally for the nonsingular holomorphic mapping GB defined by

it follows that GB 0 FoGAl has the desired form, thus concluding the proof. 9. DEFINITION. A complex submanifold M of an open subset D ~ en is a relatively closed subset M ~ D such that for every point A E M there exist an open neighborhood VA of A and a biholomorphic mapping F: VA --. .::\(0; R) onto an open polydisc in en such that F(A) = 0 and that

for some integer k.

If M ~ D is a submanifold as above, then the inverse of the restriction FI VA n M is a nonsingular holomorphic mapping

such that the image of G is precisely the subset VA n M; conversely, if for each point A E M there exists such a mapping G, then it follows immediately from Theorem 8 that M is a complex submanifold. If Gl and G2 are two such mappings, then the composition Gil 0 G2 is a biholomorphic mapping; hence, the integer k is independent of the choices of the mapping F or G. This integer is called the (complex) dimension of the submanifold M at the point A. It is clear that the dimension is locally constant and hence is a constant on each connected component M j of the set M; that constant is called simply the (complex) dimension of M;. If all connected components of the submanifold M have the same dimension k, then M is said to be of pure dimension k. The word complex is usually dropped when-

C Holomorphic Mappings and Complex Manifolds

23

ever there is no danger of confusion; a submanifold of complex dimension n is a topological space of dimension 2n. With this terminology Theorem 8 can be restated as follows. 10. COROLLARY. If F is a holomorphic mapping from an open domain D if rank JF(Z) = k for all points ZED, then:

£;

en into em and

(i) For any point A ED the set {Z ED: F(Z) = F(A)} is a complex submanifold of D of pure dimension n - k. (ii) For any point A ED there exist open neighborhoods UA of A and UB of B = F(A) such that F(UA ) is a complex submanifold of UB of pure dimension k. Proof. The local assertions are immediate consequences of Theorem 8. The set {Z ED: F(Z) = F(A)} is necessarily closed, thus providing the global assertion (i), but the image F(D) need not be a closed subset of em, so (ii) remains a local assertion.

When a complex submanifold M £; D is viewed as a set in its own right, rather than merely as a subset of D, it is clear that M looks very much like the space ek locally; that observation can be made more precise rather easily as follows. Recall that a Hausdorff topological space M is called a topological manifold of dimension d if each point A E M has an open neighborhood homeomorphic to an open subset of jRd. A coordinate covering {U,., F,.} of such a manifold is a covering of M by open sets U,. together with homeomorphisms F,.: U,. .... v.. from U,. to open subsets v.. £; jRd; the sets U,. are called coordinate neighborhoods and the mappings F,. coordinate mappings. Note that on each nonempty intersection U,. n Up of coordinate neighborhoods there are given two homeomorphisms into jRd; the compositions

are called the coordinate transition mappings for the given coordinate covering and are mappings between open subsets of jRd. The manifold M is completely determined by the subsets v.. £; jRd and the coordinate transition mappings F,.p, for M can be obtained from the disjoint union of the sets v.. by identifying a point A E v.. and a point BE Vp whenever A = F,.p(B). Note that the union of any two coordinate coverings is again a coordinate covering, but the union has more coordinate transition mappings than merely the totality of coordinate transition mappings from the two separate coordinate coverings. If d = 2n, the space jR2n can be identified with en, and the images F,.(U,.) = v.. of the coordinate neighborhoods can be viewed as subsets of en. The coordinate covering {U,., F,.} is called holomorphic if all of the coordinate transition mappings are holomorphic maps. Two holomorphic coordinate coverings are then called equivalent iftheir union is again a holomorphic coordinate covering. It is necessary to convince oneself that this is a nontrivial relation, but it is easy to see that it is actually an equivalence relation since the composition of two holomorphic mappings is again holomorphic. An equivalence class of holomorphic coordinate coverings is called a holomorphic structure, and

24

Volume I

Function Theory

a topological manifold together with a fixed holomorphic structure is called a complex manifold of (complex) dimension n. Properly speaking, the dimension is only defined on each connected component of a complex manifold, but the manifold is of pure dimension n if all connected components have the same dimension. It is a useful exercise to demonstrate that an analytic submanifold M of an open subset D s en has a natural structure as a complex manifold. Note that the notion of a holomorphic function on a complex manifold or of a holomorphic mapping between two complex manifolds can be introduced in the obvious manner. It should be observed that the same construction can be used to impose other structures on topological manifolds. For example, a coordinate covering {UIX ' FIX} of a topological manifold is said to be of class COO if all the coordinate mappings are COO mappings, and two such coordinate coverings are called equivalent if their union is again a Coo coordinate covering; this is an equivalence relation, since the composition of two Coo mappings is again a Coo mapping. An equivalence class of Coo coordinate coverings is called a Coo structure, and a topological manifold together with a fixed Coo structure is a Coo manifold. Since holomorphic functions are, of course, Coo functions of the underlying real coordinates, any holomorphic coordinate covering is necessarily a Coo coordinate covering, and any two equivalent holomorphic coordinate coverings are equivalent when viewed as Coo coordinate coverings. Thus, a complex manifold can also be viewed merely as a Coo manifold by ignoring some of the structure; or equivalently, a complex manifold can be viewed as a Coo manifold with an additional structure. It is, of course, possible that two distinct complex manifolds determine the same Coo manifold; indeed, a Coo manifold can carry a number of distinct complex structures. The more detailed study of complex manifolds is a fascinating subject on which there is a very extensive literature, but it leads in directions that somewhat diverge from those to be pursued in this book and so will not be carried out any further here. However, it is useful to introduce the following additional general construction involving complex manifolds. A regular holomorphic family of vector spaces over a complex manifold M is a complex manifold E together with a holomorphic mapping F: E --+ M and the structure of a finite-dimensional complex vector space on the subset F- 1 (Z) s E for each point Z E M. The mapping F is called the projection, and the subset E z = F- 1 (Z) s E is called the fibre over the point Z. Often a holomorphic family of vector spaces F: E --+ Mover M is denoted merely by E, with the mapping F understood. Note that for any open subset U s M the inverse image F- 1 (U) s E with the restricted mapping FIF- 1 (U): F-l(U) --+ U is a holomorphic family of vector spaces over U; this is called the restriction of the family E to U and is denoted by EI U. The simplest example of a holomorphic family of vector spaces over M is the product manifold E = M x Cr , with the natural projection F: M x IC' --+ M and the obvious structure of a complex vector space on each fibre Z x Cr ; this is called the trivial family of vector spaces over M. A homomorphism from one holomorphic family of vector spaces F: E --+ Mover M to another F' := E' --+ M is a holomorphic mapping H: E --+ E' such that F' H = F and such that the restriction HIE z : E z --+ E~ is a linear mapping for every point

C Holomorphic Mappings and Complex Manifolds

25

Z E M. An isomorphism is a homomorphism that admits an inverse homomorphism. A holomorphic vector bundle over a complex manifold M is a regular hoi om orphic family of vector spaces F: E -+ M that is locally trivial, in the sense that each point Z E M has an open neighborhood Uz such that the restriction EI UZ is isomorphic to the trivial family of vector spaces over Uz . For a holomorphic vector bundle E over M it is clear that the dimension of the fibre E z is constant as a function of Z on each connected component of M; that constant is called the rank of the vector bundle. A holomorphic vector bundle of rank one is also called a holomorphic line bundle. If F: E -+ M is a holomorphic vector bundle of rank r over an n-dimensional complex manifold M, there is a coordinate covering {Ua, G~} of the complex manifold M for which the restrictions EI Ua are locally trivial and hence for which there are isomorphisms G;: F- 1 (Ua) -+ Ua x C'. The compositions Ga = (G~ x id) 0 G;, where G~: Ua -+ Y,. are the coordinate mappings on M and id: C' -+ C' is the identity mapping, are biholomorphic mappings Ga : F-l(Ua ) -+ Y,. x Cr ~ C· X C', so that {F- 1 (Ua ), Gil} is a holomorphic coordinate covering of the complex manifold E. This coordinate covering closely reflects the vector bundle structure on E; indeed, it is clear that the images of the coordinate neighborhoods are the Cartesian products Ga (Fa- 1 (Ua » = Y,. x C' and that the coordinate transition mappings GaP = Ga 0 Gpl preserve this product structure and are nonsingular linear mappings on C' depending holomorphically on the coordinates in Y,.. Thus, these coordinate transition mappings are of the form Gap(Zp, Wp) = (G~p(Zp), H~p(Zp)' Wp), where Zp E Vp, Wp is a point of Cr viewed as a column vector, G~p are the coordinate transition functions on M, H~p: Gp(Ua n Up) -+ GL(r, C) are holomorphic matrixvalued functions defined on the appropriate subsets of Vp, and H~p(Zp)' Wp is the usual matrix product of the matrices H~p(Zp) and Wp. Such a coordinate covering is called a holomorphic coordinate bundle representing the holomorphic vector bundle E. Of course, in part a holomorphic coordinate bundle is just a holomorphic coordinate covering of the manifold M and does not involve the holomorphic vector bundle at all. Having been given the complex manifold M in terms of this coordinate covering, though, the bundle is fully described by the holomorphic mappings H~p; it is easy to verify that any holomorphic mappings H~p satisfying the natural consistency conditions in turn determine a holomorphic coordinate bundle. The consistency conditions merely amount to the conditions that GaP 0 Gpy = Gay. These conditions are most easily described in terms of the mappings Hap = H~p 0 Gp : Ua n Up --+ GL(r, C); these are holomorphic mappings from the subsets Ua n Up ~ M into the matrix group GL(r, C), and the consistency conditions are merely that (5)

Thus, a holomorphic coordinate bundle of rank rover M can be thought of as a collection of hoi omorphic mappings Hap: Ua n Up -+ GL(r, C) for some coordinate covering {Ua } of M, such that these mappings satisfy (5). Of course, there are many

26

Volume I

Function Theory

holomorphic coordinate bundles representing a given holomorphic vector bundle E corresponding to the variety of choices of coordinates it is possible to make. For a fixed coordinate covering {U",} of M, though, it is a simple matter to verify that two collections of hoi om orphic mappings H",p: U", n Up ~ GL(r, q and K",p: U", n Up ~ GL(r, q represent the same holomorphic vector bundle precisely when there are holomorphic mappings L",: U", ~ GL(r, q such that K",p(Z) = L",(Z)' H",p(Z) . Lp(Zrl whenever Z E U", n Up, where Lp(Zrl here denotes the inverse of the matrix Lp(Z). Example. A derivation at a point A of an n-dimensional complex manifold is a mapping D that associates to any hoi om orphic function f in any open neighborhood of the point A a complex constant D(f) and that has the following properties: (i) D is a local operator, in the sense that iff and 9 agree on some open neighborhood of A, then D(f) = D(g); thus, the particular neighborhoods of A on which the functions are hoi om orphic are immaterial; (ii) D is a linear operator, in the sense that if f and 9 are holomorphic in some open neighborhood of A and a and bare complex constants, then D(af + bg) = aD(f) + bD(g); and (iii) D is a differential operator, in the sense that if f and 9 are hoi om orphic in some open neighborhood of A, then D(fg) = f(A)D(g) + g(A)D(f); in particular, therefore, D(c) = 0 for any constant c. The set of all derivations at A clearly form a complex vector space; this vector space is called the complex tangent space to the complex manifold M at the point A and is denoted by TA(M). It is easy to see as follows that the set T(M) = UAeM TA(M) has the natural structure ofa holomorphic vector bundle of rank n over M; this bundle is called the complex tangent bundle to the complex manifold M. Indeed, if Z 1, •.• , z. are the components of a hoi om orphic coordinate mapping in a coordinate neighborhood U of a point A E M, and if that mapping takes the point A to the origin in en and the set U to an open neighborhood V of the origin in C·, then a holomorphic function f near the point A on M can be viewed merely as a holomorphic function f(Zl"'" z.) of n complex variables in V. This function will have a power series expansion at the origin that can be written in the form

where Jj are holomorphic functions for which Jj(O) = O. If D is any derivation at the point A, then it follows immediately from the defining properties of a derivation that (6)

where D(z}) =

tj;

conversely, it is clear that the linear differential operator (6) for any

C Holomorphic Mappings and Complex Manifolds

27

complex constants tj is a derivation at the point A, indeed is a derivation with the property that D(z) = tj • Thus the tangent space 1A(M) is an n-dimensional complex vector space, and the subset UZEU Tz(M) £: T(M) is homeomorphic to V x en under the mapping that associates to the point (Z, T) E V X en the derivation Lj=l tp}joz). Moreover, if {U«, F«} is a hoi om orphic coordinate covering of M with coordinate transition mappings F«p, then to any point Z E U« n Up can be associated the Jacobian matrix T,.p = JF.,(Fp(Z)) of the coordinate transition mapping F«p, and it is an immediate consequence of the chain rule for differentiation that these matrices satisfy the consistency condition (5) and hence determine a holomorphic coordinate bundle over M. It is then a simple matter to verify that this coordinate bundle represents the hoi om orphic vector bundle T(M) with the local trivialization as described. One final general property of holomorphic mappings and the significance of this property for complex manifolds should also be mentioned here. A holomorphic mapping F between two open subsets of en can be viewed as a differentiable, indeed even as a real analytic, mapping between those open subsets considered as subsets of ~2n. If Zj = Xj + iYj are the coordinates in en and Wj = uj + iVj are the component functions of the mapping F, there are both the n x n holomorphic Jacobian matrix JF = o(W)jo(Z) and the 2n x 2n real Jacobian matrix o(U, V)jo(X, Y). The latter can conveniently be viewed as composed of four n x n matrix blocks in the form O(U) o(U, V) o(X, Y)

( o(X)

=

o(U) ) o(Y)

o (V)

o(V)

o(X)

o(Y)

where o(U) _ o(X) -

{OU;. ox 1 =< I,..] O}; hence, j(Z) = f(Z) in U - M, and the proof is thereby concluded. The preceding theorem is trivial when n = 1, since in that case the set M must be empty; however, whenever n> 1, the theorem shows that all functions hoi om orphic in D - M, without any further assumption of local boundedness, extend to holomorphic functions in all of M. That is another example of the phenomenon of holomorphic extension already discussed in section B. One simple corollary worth noting specifically is the following.

5. COROLLARY.

An isolated singularity of a holomorphic function of more than one variable is a removable singularity.

Proof. A point is a complex submanifold of dimension zero, so Theorem 4 applies whenever 0 ~ n - 2 to show that any function holomorphic in the complement of a point extends to a holomorphic function in a full neighborhood of the point, as desired.

The use made of the Cauchy integral formula in the proof of Theorem 4 can be modified to provide a somewhat broader range of analytic extension theorems.

To simplify the notation, for a product decomposition en = en' x en" a point

D Holomorphic Extension

Z E en is as usual written as Z = (Z', Z") and a polydisc ~(A; R) spondingly written as a product ~(A; R) = ~(A'; R') x ~(A"; R").

6. THEOREM (continuity theorem).

if L\(A; S) c

~(A; R) and

£::

33

en is corre-

en en' en",

For a nontrivial product decomposition = x is a holomorphic function in an open neighborhood of

if f

the closed set

i5 =

{L\(A'; R') x L\(A"; S")}

U

([L\(A'; R') - ~(A'; S')] x L\(A"; R")}

then there is a unique holomorphic function j in ~(A; R) such that j(Z) whenever ZED.

Proof.

= f(Z)

Consider the function j defined in ~(A; R) by

noting that it is holomorphic throughout ~(A; R); indeed, whenever lej - ajl = rj for 1 ~ j ~ n' and Z" E ~(A"; R"), then f is holomorphic in an open neighborhood of ((1' ... , (n" Z") and hence j is holomorphic as desired. On the other hand, whenever Z" E ~(A"; S"), then f((l' ... , (n" Z") as a function of the variables (1' ... , (n' is hoi om orphic throughout ~(A'; R') by hypothesis and hence f coincides with j in that region. It then follows from the identity theorem that f coincides with j throughout the domain D. That suffices to conclude the proof. When A is the origin, the set i5 is evidently the closure of a Reinhardt domain D, and in general D is just the translate of a Reinhardt domain. To clarify the geometric picture, note that when A is the origin and n = 2, the base B of this Reinhardt domain is the set sketched in Figure 2. The assertion of the theorem in this case is that any holomorphic function in an open neighborhood of the closure of the Reinhardt domain D with base B necessarily extends to a holomorphic function in an open neighborhood of the closure of D, where D is the smallest

B

Figure 2

34

Volume I

Function Theory

complete Reinhardt domain containing D or equivalently is the Reinhardt domain with base the full rectangle jj = [0, r 1 ] x [0, r 2 ] £; ~2. The result thus differs from the earlier analytic extension property of Reinhardt domains described in Corollary B3. This theorem is traditionally known as the continuity theorem because it was originally formulated by Hartogs in the following form: If f is holomorphic in an open neighborhood of the annulus (A(A'; R') - ~(A'; S'» x A" and is also holomorphic in open neighborhoods ofthe polydiscs A(A'; R') x A; for some points A; such that A; -+ A" as t -+ 0, then it is necessarily holomorphic in an open neighborhood of the full polydisc ~(A'; R') x A". Sometimes it is even easier to visualize the application of the theorem in this form. The theorem can quite evidently be reformulated in terms of open subsets of en rather than of closed subsets, and it can be generalized in fairly obvious ways. If D is the interior of the set jj described in Theorem 6 and F: ~(A; R) -+ E is a biholomorphic mapping between the polydisc ~(A; R) £; en and an open subset E £; en, then the set F(D) = E is sometimes called a Hartogs figure with completion E. The continuity theorem then implies that any function holomorphic in a Hartogs figure E extends to a holomorphic function in the completion E. A consideration of the role of the Cauchy integral formula in the preceding proof does illustrate one reason for the striking differences in properties of analytic extension for functions of one and functions of several variables. In one complex variable the Cauchy integral formula expresses the value of a holomorphic function at an interior point of a domain in terms of an integral over the full boundary of the domain. The integrand has a singularity at each boundary point, so such an expression would not in general be expected to lead to an analytic continuation across the boundary. In n > 1 complex variables the Cauchy integral formula expresses the value of a hoi om orphic function at an interior point of a polydisc in terms of an integral over a proper subset of the boundary of the polydisc-indeed, over the n-dimensional set consisting ofthe product ofthe n boundary circles rather than over the full (2n - I)-dimensional boundary. The integrand has a singularity at each point of the domain of integration but that need not, and as the preceding results show sometimes does not, preclude the possibility of an analytic continuation of every holomorphic function across some part of the boundary. The continuity theorem provides a general conceptual framework for using this method to exhibit for some domains D larger domains jj to which every holomorphic function in D can be extended. Unfortunately, however, this method does rely on a good deal of ingenuity in any particular case. There is as yet no generally effective method for finding a maximal domain to which all functions can be extended. Some examples of the application ofthe continuity theorem are perhaps worth examining here. The first of these is in a sense an extension of Theorem 4. 7. COROLLARY. If M is a connected complex submanifold of dimension n - 1 in an open subset D £; en, iff is a holomorphic function on D - M, and if f can be extended as a holomorphic function through at least one point of M, then f extends to a holomorphic function in all of D. Proof. Let E be the subset of M consisting of those points through which the function f extends. Thus, A E E £; M precisely when there exist an open neighborhood VA of A in D and a holomorphic function J.". in VA such that fA (Z) = f(Z) whenever

D Holomorphic Extension

35

Z E UA n (D - M). It is clear that E is an open subset of M and by hypothesis E is nonempty; so since M is connected, in order to conclude the proof it is enough to show that E is a closed subset of M. For this purpose consider a point A E E c M and choose an open neighborhood UA of A in D and a biholomorphic mapping F: UA -+ &(0; R) such that F(A) = 0 and F(UA n M) = {Z E &(0; R): Z1 = O}. Since A E E, there are points BEE arbitrarily near A; choose such a point BEE for which F(B) = C E &(0; R/2). Now it is clear that the functionf 0 F- 1 is holomorphic intheopensubsetofthepolydisc&(C; R/2) = &(0; rd2) x &(C"; R"/2) s;;; e 1 x e n - 1 of the form {&(O; rd2) x &(C"; S")} u {(&(O; rd2) - &(0; s')) x MC"; R"/2)} for a sufficiently small polyradius S. It then follows from the continuity theorem that f 0 F- 1 extends to a holomorphic function in the full polydisc &(C; R/2). Since F(A) E &(C; R/2), that yields an extension ofthe functionf to a holomorphic function at A, showing that A E E. Thus, E is a closed subset of M, and the proof is thereby concluded.

8. COROLLARY.

Every function that is holomorphic in an open neighborhood of the boundary of a polydisc in en for n > 1 has a holomorphic extension to the full polydisc.

Proof. With the decomposition en = e 1 x en- 1 for n > 1, any point Z E en will be written Z = (z', Z") where z' = Z1 and Z" = (Z2"'" zn), and correspondingly any polydisc &(A; R) will be decomposed as a product &(A; R) = &(A'; R') x &(A"; R"). If f is holomorphic in an open neighborhood of the boundary of &(A; R), then for any fixed point Z" E .1(A"; R") the function f(z', Z") as a function of z' alone is holomorphic in an open neighborhood of the circle {z' E e 1 : Iz' - a'i = r'}, and if Z" is in an open neighborhood of the boundary of &(A"; R"), then f(z', Z") is holomorphic in the full disc &(a'; r'). It follows immediately from the continuity theorem, as Z" moves from the boundary of &(A"; R") to its interior, that f(z'; Z") extends to a holomorphic function in an open neighborhood of each disc &(a'; R') x Z", and hence f has a holomorphic extension to &(A; R) as desired.

9. COROLLARY. Suppose that g1 and g2 are holomorphic functions in an open subset D s;;; en and that they describe a nonsingular holomorphic mapping from D into e 2 • If A ED is a point in D at which gl (A) = g2(A) = 1, introduce the open subset

U = {Z ED: Igl(Z)1 > I} u {Z ED: Ig2(Z)1 > I} cD which has A as a boundary point. Then any function that is holomorphic in U extends to a function that is holomorphic in an open neighborhood of A. Proof. The corollary is a local result, so D can be replaced by any sufficiently small open neighborhood ofthe point A. Choose functions g3"'" gn that are hoi om orphic in D, vanish at A, and are such that the mapping G: D -+ en with coordinate functions 9 l ' ... , g" is nonsingular at A. Then by the inverse mapping theorem G is a biholomorphic mapping from an open neighborhood of A into en. When this mapping is viewed merely as a change of coordinates near A in Cn , it clearly

36

Volume I

Function Theory

suffices to prove the corollary for the special case in which A = (1, 1,0, ... ,0). In this special case

g1 = Z1' g2 = Z2'

and

where D is some open neighborhood of A. Now introduce the further nonsingular change of coordinates

that takes the origin to the point A, and let D* and U* be the inverse images of the open sets D and U under this change of coordinates. The problem is to show that any function that is holomorphic in u* extends to a function that is holomorphic in an open neighborhood of the origin W = 0. Note that the point (0, W2' 0, ... , 0) corresponds to the point (Z1' Z2' 0, ... ,0) where Z1 = 1 + W2/2 and Z2 = 1 - w2/2. Here

where W2 = U2 + iV2. Ifu 2 > 0, then IZ11 > 1; ifu 2 < 0, then IZ21 > 1; and ifU2 = and V 2 ~ 0, then IZ 11 > 1 and IZ 21 > 1. Thus, U* contains the annulus

°

if e is chosen small enough that this set is contained in the open neighborhood D* of the origin. On the other hand, for any positive real number r, the point (2r, W 2 , 0, ... ,0) corresponds to the point (Z1' Z2' 0, ... ,0) where Z1 = 1 + r + W2/2 and Z2 = 1 + r - w2/2. Here

If U2

~

0, then Iz 11 > 1, and if U2

~

0, then Iz 21 > 1. Thus, U* contains the disc

if r is chosen small enough that this set is contained in the open neighborhood D* of the origin. As r approaches zero, the discs Ar approach the disc Ao containing the origin. So it follows from the continuity theorem that any function that is holomorphic in U*, hence in an open neighborhood of Ar for r > 0, extends to a function that is holomorphic in an open neighborhood of Ao, hence in an open neighborhood of the origin, as desired to conclude the proof. The equations Igi(Z)1 2 = 1 define smooth real hypersurfaces Hi in the Euclidean space ~2n in an open neighborhood of the point A. Since

D Holomorphic Extension

37

it follows that the gradients of the real functions Ig;l2 are linearly independent vectors at the point A, and hence the hypersurfaces H; meet transversally at the point A. These two hypersurfaces thus decompose an open neighborhood of the point A into four quadrants, and U is the union of three of them. The assertion of the corollary is that any function that is holomorphic in U extends to a function that is holomorphic in a full open neighborhood of the corner point A. For that reason this assertion is sometimes called the "corner theorem." The corollary holds for any three of the four quadrants, not just for U alone, for the result is a local one and the corollary can be applied to the function l/g; near the point A. To discuss the next example ofthe application ofthe continuity theorem, it is convenient to introduce some standard and generally useful terminology. 10. DEFINITION. A tube domain D £; en with base B D = {Z E en: Re Z E B}.

£;

IRnis an open subset of en of the form

If en is viewed as the direct sum en = IRn + ilR n of the real subspace and the purely imaginary subspace, a tube domain D £; en with base B £; IR n can be expressed as a subset of en of the form D = B + mn. It is clear that tube domains can be characterized as those open subsets of en that are mapped to themselves by any translation by a purely imaginary vector-that is, as those open subsets D £; en such that D = D + iX for any vector X E IRn. From this in turn it is evident that if all functions that are holomorphic in a tube domain D £; en can be extended to holomorphic functions in a properly larger open subset of en, then indeed all holomorphic functions in D can be extended to holomorphic functions in a properly larger tube domain in en. Pairs of tube domains for which such holomorphic extension is possible can be described very simply. To begin with a suggestive special case, which will then be used to derive the general result, consider a tube domain D £; en, n ~ 2, that has as base a connected open neighborhood B ofthe union ofthe unit intervals {(u I, 0, 0, ... , 0): ~ U 1 ~ I} and {(O, u 2 , 0, ... ,0): ~ U2 ~ I} on the first two coordinate axes in IRn, and introduce the closed subset K = H + mn where H = {(u l , U2' 0, ... ,0): U 1 ~ 0, U 2 ~ 0, U 1 + U2 ~ I} is the convex hull of the same union of unit intervals. An example of this configuration in the special case n = 2 is sketched in Figure 3.

°

D=B+ilR" K=H+ilR"

--_~... '(I.O)

Figure 3

°

38

Volume I

Function Theory

11. LEMMA.

There exists an open neighborhood U of the closed set K such that any function holomorphic in D can be extended to a function holomorphic in D u U.

Proof. In order to prove the desired result, it suffices to show that for any point WE K there exists a biholomorphic mapping F: .1(0; R) -+ Uw between an open polydisc .1(0; R) ~ en and an open subset Uw ~ en containing the point W, such that F(Z) E D if either IZll < r 1 , IZ21 < S2, ... , IZnl < Sn or SI < IztI < r 1 , IZ21
O} for some radius r > 1, as sketched in Figure 4. To be more precise, the particular conformal mapping f of interest is that for which f(1) = 1,J( -1) = 0, andf( ± i) = ± ia E (3L\(1 - r; r). In Figure 4 note thatthe angle {} is the same as the angle between the two boundary arcs of E at the points ± ia, and from elementary geometry r- 1 = 2 sin2({}j2) and a = cot({}j2). In terms of these parameters the conformal mapping f can be constructed as the composition f(z) = f3(f2(fl (z))), where fl (z) = (i + z)j(i - z) takes .1(0; 1) to the right half-plane, f2(Z) = zlJ/1t takes the right half-plane to the sector lying between the two rays from the origin making angles (}j2 with the real axis, and f3(Z) = ia

z - e ilJ /2 "1J/2 z + e'

Imw

Imz

f

Rez

-\

-i

Figure 4

Rew

o

Holomorphic Extension

39

takes this sector to E as desired. From these explicit formulas it follows that

J( -z)

1 - J(z) 1 + a-2J(z)

=

(4)

that is, the automorphism of the disc ~(O; 1) sending z to - z is transformed by J to the automorphism (4) of the region E, with both automorphisms being elliptic linear fractional transformations of order two. Now introduce the holomorphic mapping G: ~(O; 1) x C* X cn- 2 _ cn defined by G(z, t,

Z3' ... ,

zn) = (tf(z), tf( - z)

+ 25ait, Z3' ••. , zn)

(5)

where C* denotes the complement of the origin in C. It is a straightforward matter to verify that G is a biholomorphic mapping between ~(O; 1) x C* X cn-2 and its image whenever a> 2. Indeed, the determinant of the Jacobian matrix of the mapping G is det JG(z, t, Z3'

... ,

zn) = tf ,(z) [ 25az. + J( - z) + J(z) 1'(-Z)] 1'(z)

It follows from (4) that

1'(

1

-Z)I

1'(z)

whenever z E

1

= (1

~(O;

Idet JG(z, t,

1 + a- 2 1 + a-2J(z))2
2, since then J(z)

Z3' ... ,

zn)1

~

E

E and hence IJ(z) I < a. Therefore,

Itf'(z)I(25a - a - 5a) > 0

so the mapping G is nonsingular in the specified domain. To see that G is biholomorphic, it is then only necessary to show that distinct points in the domain of G have distinct images. Suppose to the contrary that G(z', t', z;, ... , z~) = G(z", t", zi, ... , z:) for two distinct points (z', t', z;, ... , z~) and (z", t", zi, ... , z:) in A(O; 1) x C* X cn- 2 • It is evident from the form of the mapping G that t' oF t" and z' oF z" but that z; = zi, ... , z~ = z:. Furthermore, setting A. = t'/t" and, = J(z') and using (4) again, show that (6)

where' E E, A., E E, and A. oF 1. If a > 2, then for any fixed point' E E the right-hand side of (6) is a holomorphic function of A. in an open neighborhood of the disc IA. - 11 ~ t, since lei < a and the expression in braces vanishes at A. = 1. Therefore, iflA. - 11 ~ t, the absolute value ofthe right-hand side of (6) is bounded from above by its maximal absolute value on the circle IA. - 11 = t, so that

40

Volume I

Function Theory 25a ~ 2

sup

1),-11=1/2

I1 +I-A' a

2,y 1\"

' A 1 --2 1+a

,

I< 18a + 14 < 25a

a contradiction. On the other hand, if IA-II ~ !, then since' E E and A' that 1'1 < a and lAC! < a, a straightforward estimation based on (6) yields 25a

< -I 1 11' {2(1 + a) + IAI2(1 + a)} I\, -

E

E, so

~ 2(1 + a) 1),-11 2, again a contradiction. That shows that G is biholomorphic as desired. Since Re G(z, 0, 0, ... ,0) = 0 E B for all z E ~(O; 1), it follows from continuity alone that there are positive constants ~, r3' ... , rn such that G(z, t, Z3' ... , zn) E D iflzl < 1, It I 0, and the theorem holds trivially in that case. For the induction step write the differential form ¢J as ¢J = dZk 1\ (1. + p, where (1. and p are differential forms that involve only the conjugate differentials dz l , ... , dZk- l . Since = a¢J = - dZk 1\ a(1. + ap, it is clear that the forms (1. and pare holomorphic in the variables Zk+l' ••. , Zno for each separate coefficient must be annihilated by the operators OjOZk+l' ... , ojozn. Any coefficient f in the explicit representation of the differential form (1. can be view~d as a Coo function of the single variable Zk in an open neighborhood of the disc L\(ak; rk), and f is also a Coo function of the auxiliary parameters Z 1, ••• , Zk-l in L\(al' ... , ak - l ; r l , ... , rk- l ) and a holomorphic function of the auxiliary parameters Zk+l' •.• , Zn in L\(ak+l' ... , an; rk+l' .•• , rn). It follows from Lemma 2 that there exists a Coo function 9 in an open neighborhood of L\(A; R) such that OgjOZk = f and that 9 is holomorphic in the variables Zk+l' ... , Zn' If y is the differential form obtained from (1. by replacing_each coefficient f by the corresponding coefficient g, it then clearly follows that oy = ~ + dZk 1\ (1., where (j is a differential form involving only the conjugate differentials dz l , ... , dZk - 1 • Next set () = ¢J = p - (j, and note

°

ay

46

Volume I

Function Theory

that einvolves only the conjugate differentials dz l ' ... , dzk - 1 , since f3 and b involve only these differentials, and that ae = af/J - aay = o. So by the induction hypothesis there exists a Coo differential form" in an open neighborhood of K(A; R) such that e = a". Finally set IjI = " + y; it follows immediately that aljl = f/J, and that suffices to conclude the proof. It is, of course, an immediate consequence of Dolbeault's lemma that if f/J E Sb,q for q > 0 and if af/J = 0, then in some open neighborhood VA of each point A E D there exists a differential form IjIA such that aljlA = f/JI VA- Thus, the condition that af/J = 0 is at least locally a necessary and sufficient condition for the existence of a solution IjI to the partial differential equation aljl = f/J. The condition that af/J = 0 is necessary but generally not sufficient for the global existence of a solution IjI to the partial differential equation aljl = f/J-that is to say, for the existence of a differential form IjI E Sb,q-l such that aljl = f/J. Actually the global integrability of this differential equation involves deeper complex analytic properties of the domain D, and the study of this differential equation is one of the principal themes in function theory in several variables. Although the study of this differential equation is only begun here with the discussion of a special case and some illustrative applications, it is convenient to introduce the basic terminology and notation from the outset.

A differential form f/J E Sb,q is a-closed if af/J = 0, and it is a-exact if there exists a differential form IjI E Sb,q-l such that f/J = aljl. Every a-exact form is a-closed, since aa = 0, and the quotient complex vector spaces

4. DEFINITION.

,tt'p,q(D) = {a-closed forms of bidegree (p, q) in D} {a-exact forms of bidegree (p, q) in D} are the Dolbeault cohomology groups of the domain D. The

aoperators can be viewed as a sequence of linear mappings a

a

a

a

(4)

Sb'o ----+ Sb' 1 ----+ Sb' 2 ----+ ... ----+ Sb' n

such that aa = O. The a-closed forms are the kernels of these linear mappings, while =0 the a-exact forms are the images of these linear mappings. The condition that is just that the image of one of these linear mappings is contained in the kernel of the next linear mapping, and the Dolbeault cohomology groups measure the extent to which the image of one of these linear mappings falls short of coinciding with the kernel of the next linear mapping. Cohomology groups arising in similar ways from other sequences of vector spaces and linear mappings will be considered in more detail in Volume III. The beginning and end ofthe sequence (4) are somewhat special in this regard. The a-closed (p, O)-forms are precisely those forms of bidegree (p, 0) having all coefficients holomorphic; they are called the holomorphic p-forms in D and form a linear subspace of 8b'o denoted by (9[,,0. In particular, of course,

aa

E The "0 Operator

=

47

is just the algebra of holomorphic functions in D. There are no 8-exact = (!)Jj'o. On the other hand, all (p, n)-forms are 8-closed, so ,}f"p,n(D) = rffb' n/8rffb' n-1 , and trivially ,}f"p,q(D) = 0 whenever q > n.

(!)g'o

(!)D

(p, O)-forms, so ,}f"p,O(D)

5. THEOREM. If D q > O.

~

C" is an open polydisc in the extended sense, then ,}f"p,q = 0 whenever

Proof. Let Dr be a sequence of concentric open polydiscs in C" such that Dr ~ Dr+1 and that UrDr = D, and consider any 8-closed differential form fjJ E rffb,q for q > O. The theorem will be proved by constructing a form 1/1 E rffb,q-1 such that 81/1 = fjJ, the construction proceeding inductively over the compact approximating subsets Dr ~ D. The cases q = 1 and q > 1 require separate treatment, however.

i. If q > 1, construct by induction on r a sequence of (p, q - I)-forms I/Ir such that I/Ir is a Coo form in an open neighborhood of D" 81/1r = fjJ in that neighborhood, and I/Ir+11 Dr = I/Ir; then 1/1 E rffb,q-1 defined by 1/1 IDr = I/IrlDr is the desired form. The existence of 1/11 follows immediately from Dolbeault's lemma. For the induction step, suppose that 1/11' ... , I/Ir have already been constructed as desired. It follows from Dolbeault's lemma that there exists a (p, q - I)-form 1/1;+1 that is Coo in an open neighborhood of Dr+1 and satisfies 81/1;+1 = fjJ in that neighborhood. The difference 1/1;+1 - I/Ir is then a 8-closed (p, q - I)-form in an open neighborhood of D" and since q - 1 > 0 it follows from Dolbeault's lemma again that there exists a (p, q - 2)-form '1r in an open neighborhood Ur of Dr such that 8'1r = 1/1;+1 - I/Ir in that neighborhood. Choose a Coo function Pr in C" such that Pr(Z) = 1 for all points Z in an open neighborhood of Dr and Pr(Z) = 0 in an open neighborhood of D - Ur ; the existence of such a function is a well-known result. The product Pr'1r clearly extends to a Coo differential form throughout D that vanishes in D - Ur, and if Pr'1r denotes this extension, set I/Ir+1 = 1/1;+1 - 8(Pr'1r) in an open neighborhood of Dr+1. This form clearly has the desired properties, and that concludes the induction step. ii. The preceding argument breaks down when q = 1, and a slightly more complicated construction is necessary in that case. Construct a sequence of (p, 0)forms I/Ir such that I/Ir is a COO form in an open neighborhood of D" 81/1r = fjJ in that neighborhood, and I/Ir+1 - I/Ir = ()r is a holomorphic p-form in an open neighborhood of Dr with all coefficients of ()r being bounded by rr throughout Dr. Then I/I(Z) = limr I/Ir(Z) is a well-defined (p, O)-form in D; and since for any r this limit form can be written as I/IIDr = I/IrlDr + Ll=r()jID" where Lj()jlDr is a uniformly convergent series of holomorphic p-forms, it follows that I/IIDr differs from I/IrlDr by a holomorphic p-form and hence that 1/1 has the desired properties. The existence of 1/11 follows immediately from Dolbeault's lemma. For the induction step, suppose that 1/11"'" I/Ir have already been constructed as desired. It follows from Dolbeault's lemma that there ~xists a (p, O)-form 1/1;+1 that is Coo in an open neighborhood of Dr+1 and satisfies al/l;+l = cp in that neighborhood. The difference (); = 1/1;+1 - I/Ir is then a holomorphic p-form in an open neighborhood of Dr. The power series expansions of the coefficients of (); about the common center of all the polydiscs are absolutely and uniformly convergent in Dr; hence, 0; = ();' + 0" where the coefficients of 0; are polynomials and the coefficients of (Jr are bounded by 2- r throughout

48

Volume I

Function Theory

Dr. The form t/lr+! defined in an open neighborhood of Dr+! by t/lr+l = t/I;+l - e;' then satisfies all the desired conditions, and that suffices to conclude the proof. Two illustrative examples indicate the usefulness of the preceding results. The first is an extension of Corollary 07. 6. THEOREM (Hartogs's extension theorem). If K is a compact subset of an open set U s en for n > 1 and if U - K is connected, then any function that is holomorphic in U - K extends to a function that is holomorphic in U. Proof It can be assumed without loss of generality that [j is also compact. Choose a Coo real-valued function p in en such that p(Z) = for all points Z in an open neighborhood of K and that p(Z) = 1 for all points Z in an open neighborhood of en - U. To any function f that is holomorphic in U - K, associate the function I' in U defined by

°

I'(Z) =

{~(Z)f(Z)

if Z if Z

E

U - K

E

K

Note that as a consequence of the properties of the function p, this function I' is Coo throughout U and is holomorphic near the boundary of U. The differential form ¢J in en defined by ¢J(Z)

=

{~'(Z)

if ZE U if Z E en

-

U

is then evidently a Coo and a-closed form ofbidegree (0, 1) throughout en. It follows from Theorem 5 that there exists a Coo function g in C" such that ¢J = ago This function g is, of course, holomorphic in an open neighborhood of en - U, since ¢J vanishes there, and it follows from Corollary 08, applied to any polydisc large enough to contain tJ, that the restriction of g to the unbounded component of en - U is a holomorphic function there that extends to a holomorphic function g' throughout all of en. Now it is possible to replace g by g - g', since a(g - g') = ag = ¢J, and hence it is possible to assume that the function g vanishes identically in an open neighborhood of the unbounded component of en - U. Under this assumption introduce the function h = I' - g in U, noting that it is holomorphic in U, since ah = ¢J - ag there. Furthermore in the intersection with U of a sufficiently small open neighborhood of any boundary point of U that is also a boundary point of the unbounded component of en - U, necessarily h = f, since I' = pI = I and g = there. Hence by the identity theorem h coincides with f throughout U - K, so h provides the desired holomorphic extension of f, and the proof is thereby concluded.

°

7. THEOREM. Let D ~ en be an open subset such that £o.l(D) = 0, let h be a nonsingular holomorphic mapping from D into e, and let M s D be the complex submanifold

E The

aOperator

49

M = {Z E D: h(Z) = OJ. Then for any function f that is holomorphic on M there exists a function g that is holomorphic in D and is such that glM = f

Proof. In an open neighborhood UA of any point A E M there exist local coordinates Z l' •.. , Zn such that A is the origin and h(Z) = Zm and thus

The restriction fl UA is just a holomorphic function of the complex variables (z 1, ... , Zn-l) in an open neighborhood of the origin in Cn - 1, and it can be extended to a holomorphic function fA in all of UA merely by setting fA (z l' ... , Zn-l' Zn) = f(z l' ... , Zn-l' 0). In this way it is possible to find a covering of the domain D by open subsets ~ and holomorphic functions fj in the sets ~, such that fjl ~ ( l M = fl ~ ( l M. The functions fj can of course be chosen quite at random if ~ ( l M = 0, and it can be assumed that the covering {~} of D is locally finite. On the nonempty intersections Ui ( l ~, the differences h - fj are holomorphic functions that vanish on M ( l Ui ( l ~,and hence the quotients hj = (h - fj)lh remain hoI om orphic. These functions clearly satisfy the conditions: whenever Z

E

Ui ( l

~

(5)

Now it is relatively easy to construct COO functions gi in the various sets

Vi such that

Indeed, simply choose a Coo partition of unity {pJ subordinate to the covering {UJ, and for any point Z E Ui set

where the summation is extended over all indices k such that Z E Uk. For any index k the function Pkhk clearly extends to a Coo function in ~, and since the sum is finite for a locally finite covering, the function gi is therefore also a Coo function. It follows from (5) and the properties of a partition of unity that for Z E Ui ( l ~,

In terms of these functions, next introduce the differential form tP in D defined by tP(Z) =

8gi(Z) whenever Z E

Ui

50

Volume I

Function Theory

noting that agj(Z) - agiZ) = afij(Z) = 0 whenever Z

E

Uj () ~,since the functions

k are holomorphic. Thus, tP is a well-defined COO and a-closed form ofbidegree (0, 1) in D. Since £,0,1 (D) = 0 by hypothesis, there exists a COO function g' in D such that og' = tP. Finally, introduce the function 9 in D defined by g(Z)

= /;(Z) -

h(Z) [gj(Z) - g'(Z)]

whenever Z

E

Uj

noting that it is a well-defined Coo function in D, since /;(Z) - h(Z)gj(Z) = jj(Z) - h(Z)gj(Z) whenever Z E Uj () ~. Moreover, ag = 0 by construction, so 9 is actually holomorphic in D, and g(Z) = /;(Z) = f(Z) whenever Z E Uj () M, since h(Z) = 0 there. Thus, 9 is the desired function, and the proof is thereby concluded. It should be noted that nowhere in the preceding proof was the condition that D be an open subset of en really used; the same conclusion thus holds whenever D is any complex manifold such that £,0,1 (D) = O. The simplest example of such a complex manifold is just an open polydisc in en, in view of Theorem 5; other examples will be discussed later. When D is an open disc in e 1, a complex submanifold is just a discrete set of points in D, and as is well known, for any such set of points there exists a function that is holomorphic in D and has simple zeros at all these points. So in this case the preceding theorem merely asserts that there exists a holomorphic function in D taking any preassigned values at any discrete set of points in D. It should be pointed out that actually the proof of Theorem 6 goes through in almost unaltered form with Theorem 3 in place of the rather stronger Theorem 5. A somewhat weaker form of Theorem 7 can also be established, using weaker hypotheses paralleling the differences between Theorem 3 and Theorem 5. Since this is of use in some circumstances, it is worth including an explicit statement here, but the necessary modification of the proof of Theorem 7 will be left as an exercise.

8. COROLLARY. Let D ~ en be an open domain with the property that for any Coo differential form tP of bidegree (0, 1) and satisfying atP = 0 in an open neighborhood of D, there exists a Coo function p in a perhaps smaller open neighborhood of D such that ap = tP there. If h is a nonsingular holomorphic mapping from an open neighborhood U of D into e and M ~ U is the complex submanifold M = {Z E U: h(Z) = O}, then for any function f that is holomorphic on M there exist an open subneighborhood V ~ U of D and a holomorphic function 9 in V such that glM () V = flM () V.

The passage from local solutions of various problems to global solutions of these problems by an auxiliary construction involving functions {k} satisfying (5) is a very common form of proof in the study of holomorphic functions of several variables, and it can already be found in the work of Cousin at the tum of the century. For that reason a collection of hoI om orphic functions {/;j} satisfying (5) is sometimes called Cousin data for the covering {Uj }. A considerable bit of effort will eventually be expended on the construction of a machine that performs such arguments rather mechanically; that effort is amply rewarded by the ease with which the machine operates when once operational. It should be pointed out, though, that

E The

"0 Operator

51

this method does not always work, indeed that the Dolbeault cohomology groups Jfp.q(D) are not always trivial for q > O.

Example. If D is the space of two complex variables with the origin deleted, then Jfo.l(D) -1= O. To show that, set r2 = IZll2 + IZ212 and note that

It is thus possible to define a a-closed form

,p E eg· l

in this domain D by setting

It is easy to see that there cannot exist any COO function 1 E eg· o such that al = ,p, and hence that ,p represents a nontrivial element of the Dolbeault group Jfo.l(D). Indeed, if al = ,p, then 9 = zd - Z2r-2 is holomorphic at all points ZED at which Z 1 -1= 0 and is locally bounded in D. Hence by the extended Riemann removable singularities theorem, 9 extends to a holomorphic function in D, but then by Theorem D4 the function 9 extends further to a function holomorphic in all of (:2. However, as Zl approaches zero, it follows for this extension that g(O, Z2) = 1/z 2 ; hence, 9 cannot even be continuous at the origin, a contradiction.

F Polynomial Approximation

A number of results about holomorphic functions can be expressed very conveniently in terms of a topology on the vector space (9D of holomorphic functions in an open subset D ~ en, and indeed it is but natural in analysis to study various topological vector spaces. It was observed in section A that if a sequence offunctions in (9D converges uniformly on compact subsets of D, then the limit function also belongs to (9D; that suggests imposing on (9D the topology for which this is the appropriate notion of convergence, the compact-open topology. 1. DEFINITION. The natural topology on (9D is the topology for which a basis of the open neighborhoods of a point f E (9D consists of subsets of (9D of the form UK •• (f)

= {g E (9D: Ig(Z) - f(Z)1
O.

A sequence of holomorphic functions on D thus converges in the natural topology of (9D precisely when these functions are uniformly convergent on any compact subset in D. It is clear that (9D is a topological vector space under the natural topology; addition and scalar multiplication are continuous mappings. It is also clear that the space (9D is Hausdorff under the natural topology. Note that basic sets of the same form can also be used to impose a topology on the vector space C(/D of continuous complex-valued functions on D. The natural topology on (9D is then the topology (9D inherits as a subset of C(/D' and (9D is a closed subspace of reD' The natural topology on (9D can also be described in terms of the following family of pseudo norms. 2. DEFINITION.

To any compact subset K

by

IIfIIK = sup If(Z) 1 ZeK

52

~ D

associate the pseudonorms

II

Ilk on (9D defined

F Polynomial Approximation

53

II 11K is a well-defined real-valued function on (!)D, and it satisfies the usual pseudonorm properties: (i) II/IIK ~ 0 for all IE (!)D; (ii) III + gilK ~ II/IIK + IlgilK for all J, g E(!)D; and (iii) Ilc/il K= Iclll/ll K for all c EC, IE (!)D' If II/IIK = 0, then I vanishes on K, and as a consequence of the identity theorem, I also vanishes on any connected component of D that meets the interior of K. So if D has finitely many connected components and the interior of K meets each component, then II/IIK = 0 implies that 1= 0; hence, II 11K is actually a norm. The natural topology on (!)D is that defined by this family of pseudonorms, since a basis for the open neighborhoods of a point I E (!)D consists of subsets of (!)D of the form It is clear that

and since these sets are obviously convex, it follows that the topological vector space is locally convex. The natural topology on (!)D can actually be defined by a countable set of these pseudo norms, for it is enough to choose countably many compact subsets K. s;; D such that the interiors ofthese subsets cover D, and merely to consider the pseudonorms II II. = I 11K,. That shows that the topological vector space (!)D satisfies the first axiom of countability, and it further shows that the natural topology on (!)D is actually a metric topology, since the expression (!)D

(1)'

p(f,g)=~ 2

III - gil. III-gil.

1+

(1)

is easily seen to be a metric describing that topology. Finally it is also clear that the vector space (!)D is complete in terms of this metric. To summarize these various properties then, (!)D with its natural topology is a Frechet space. Of course, the situation would be somewhat simpler if (!)D with its natural topology were a Banach space rather than merely a Frechet space, but that is unfortunately not the case. To see that, suppose to the contrary that (!)D with its natural topology is a Banach space under some norm I II. Any positive multiple of this norm is also a norm, so it can further be supposed that 11111 = 1. The space of all continuous linear mappings T: (!)D -+ (!)D is also a Banach space under the norm

IITII

=

sup II Tgil gefJD

Ilgll

indeed is even a Banach algebra, as is the case for the space of continuous linear endomorphisms of any Banach space. To each IE (!)D there can be associated the linear mapping 1f: (!)D -+ (!)D defined by 1f(g) = Ig, and it is clear that this is a continuous linear mapping in the natural topology of (!)D' It is also clear from the definition of the norm 111f II that

11/·111

111f1l~~=lIfll

54

Volume I

Function Theory

and hence that 11f" I ~ 111J. I = II1Jv I ~ 111Jllv for any integer v ~ 1. If 111J II < lei for some constant c E e, then the series II (flc)V II ~ 111J/c II = Icl- 111J II" is convergent; hence, the series (flc)" converges to an element g E (!)D for which (c - f). g = c. That means that (c - f)-I is a holomorphic function in D whenever IcI > 111J II, which is clearly impossible if the function f is unbounded, and this contradiction shows that (!)D cannot be a Banach space. For any subset S £; (!)D it is obvious from the preceding definitions that the closure S is a compact subset of (!)D precisely when every sequence of functions in S contains a subsequence that is uniformly convergent on every compact subset of D. Thus, the condition that S be compact is just the classical condition that S be a normal family of holomorphic functions. The traditional characterization of normal families of holomorphic functions of one variable extends immediately to several variables as follows.

Lv

Lv

Lv

v L

V

3. THEOREM (Montel's theorem). A subset S £; (!)D has compact closure in (!)D if and only if the functions in S are uniformly bounded on every compact subset of D. Proof. Since the necessity of this condition for the compactness of S is obvious, it remains only to prove the sufficiency. If all the functions in S are uniformly bounded on a closed polydisc ~(A; R) c D, then it follows immediately from Cauchy's inequalities, Theorem A5, that the partial derivatives iJfliJzj are uniformly bounded on any smaller polydisc ~(A; Rd c L\(A; R). But that implies that the functions in S are uniformly equicontinuous on ~(A; R 1 ); hence, any sequence offunctions in S contains a subsequence converging uniformly on L\(A; Rl)' Choosing a countable family of such polydiscs L\(A; Rd covering D and using the usual Cantor diagonal argument then complete the proof of the theorem.

In place of the statement that the functions in S are uniformly bounded on every compact subset of D, the equivalent statement that the functions in S are locally bounded in D is often used. That the functions in S are locally bounded in D, of course, means that to each point A E D there correspond an open neighborhood U of the point A and a constant M such that If(Z) I ~ M whenever f E Sand Z E U. For yet another reformulation of this theorem, note that if D and E are any open subsets of C" such that D £; E, then the restriction to the subset D of any function f E (!)E is clearly a function flD E (!)D' and the mapping f -+ flD is a linear mapping rDE : (!)E -+ (!)D' Recall that a linear mapping between two topological vector spaces is called compact (or completely continuous in an older terminology) if some open neighborhood of the origin in the domain is mapped to a set having compact closure in the range. 4. COROLLARY. If D and E are open subsets of en such that 15 is compact and 15 £; E, then the restriction mapping 'DE: (!)E -+ (!)D is a compact linear mapping. Proof. The set Ujj,. = {f E (!)E : If(Z)1 < e whenever Z E 15} is a basic open neighborhood of the origin in the vector space (!)D' and the restriction rDE(Ujj,.) is a subset

F Polynomial Approximation

of (!JD that has compact closure in result.

(!JD

55

by Montel's theorem; that proves the desired

Next, in a slightly different direction, the familiar theorem of Runge for holomorphic functions of one variable asserts that any holomorphic function in a simply connected open subset D!;;; 1[;1 can be approximated uniformly on any compact subset of D by a polynomial function. That too is an assertion that is easily expressed in terms of the natural topology on (!JD. 5. DEFINITION.

The subspace f!lJD !;;; (!JD is the closure in the natural topology of subspace of (!JD consisting of the restrictions to D of polynomial functions.

(!JD

of the

Equivalently, of course, f!lJD is the closure of the linear subspace r DE ( (!JE) !;;; (!JD' where E = I[;n and r DE is the restriction mapping as above. With this notation, Runge's theorem is just the assertion that f!lJD = (!JD for any simply connected open subset D !;;; 1[;1. For subsets D !;;; en with n > 1, the situation is much more complicated. There is no simple topological characterization of subsets D !;;; en for which f!lJD = (!JD; indeed, there is not even any intrinsic analytic characterization of such subsets, as will shortly be shown. Of course, f!lJD = (!JD whenever D is an open polydisc, as is obvious from consideration of the power series expansion about the center of D offunctions in (!JD' and one-variable methods can be used to extend this observation somewhat. If K = Kl X K2 C 1[;1 X I[;n-1, where Kl and K2 are compact sets and Kl is simply connected, then any function that is holomorphic in an open neighborhood of K can be approximated uniformly on K by functions of the form g(Z) = L~=o g.(Z2' ... , zn)zj', where g. are holomorphic in an open neighborhood of K 2.

6. LEMMA.

Proof. If f is holomorphic in an open neighborhood of K, then by the Cauchy integral formula

for all Z in an open neighborhood of K, where y is a suitable simple closed curve containing Kl in its interior. The denominator el - ZI is bounded away from zero for el E y and ZI E K 1 , so the integral can be approximated uniformly on K by its Riemann sums L.(e. - zd- 1f(e., Z2' ... ' zn) for some finite set of points E y. To complete the proof, it suffices merely to show that for any point a 1 ¢ Kl the function (al - zd- 1 can be approximated uniformly on Kl by polynomials in ZI. For this purpose choose a curve a(t) in 1[;1 - Kl such that a(O) = a 1 and la(t)l-+ 00 as t -+ 00; there exists such a curve, since K 1 is simply connected. The set T = {t E [0, (0): (a(t) - zd- 1 can be approximated uniformly on Kl by polynomials} is clearly a closed subset of [0, (0) and is nonempty, since it evidently contains

e.

56

Volume I Function Theory

all points t sufficiently large. On the other hand, T is also open. Indeed if to and t E [0, (0) is any point such that

la(t o) - a(t)1 < inf la(t o) Z1

E

T

Z 11

eKl

then

r

where the series converges uniformly for Z 1 E K 1. Thus, (a(t) - Z 1 1 can be approximated uniformly on K 1 by finitely many terms of this series, and each term can in turn be approximated uniformly on K 1 by polynomials since to E T, so that t E T and T is hence open. Since Tis nonempty and both open and closed, it follows that T = [0, (0); in particular, E T, and that is sufficient to conclude the proof.

°

If D = D1 connected open subsets ofCl, then 9 D = @D.

7. THEOREM (extended theorem of Runge).

X •••

x Dn £

cn where Dv are simply

Proof. It suffices to show that for any function f E @D' any compact subset KeD, and any constant e > 0, there exists a polynomial p such that If(Z) - p(Z)1 < e whenever Z E K. Furthermore, it is enough to show this when K = K 1 X ... x K n, where the sets K j are simply connected. The proof will be by induction on n. The case n = 1 was demonstrated in the preceding lemma. It also follows from the preceding lemma that any f E @D can be approximated uniformly on K by functions of the form g(Z) = L~=o gv(Z2' ... , zn)zr, where each gv is holomorphic in an open neighborhood of K2 x ... x Kn. But then by the induction hypothesis each gv can be approximated uniformly on K2 x ... x Kn by polynomials in Z2' ... ' Z"' and that clearly suffices to complete the proof. Example. The following example of a subset D c C 3 that is biholomorphic to a polydisc but for which polynomial approximation fails is due to J. Wermer.1t shows that in general there is no intrinsic analytic characterization of domains for which polynomial approximation holds. Consider the holomorphic mapping F: C 3 -+ C 3 given by (2)

noting that JF(z) = 1 - 2Z3 so that F is a local biholomorphic map whenever oF t. The mapping F is actually a biholomorphic map from the polydisc .1. = .1(0, 0, 0; 1 + e, 1 + e, e) into C 3 for some e > 0 sufficiently small. Indeed, if F is not one-to-one for any such polydisc, then there are points Av , Bv E .1(0, 0, 0; 1 + l/v, 1 + l/v, l/v) such that Av oF Bv but F(Av) = F(Bv). Subsequences of these

Z3

F Polynomial Approximation

57

points can be chosen so as to converge to some points A = (ai' a 2 , 0) and B = (b l , b2 , 0) such that A i= B but F(A) = F(B). But then from the explicit form (2) for the mapping F it follows that

Hence, A = B, a contradiction. Now choose e > 0 sufficiently small that F is a biholomorphic mapping from the polydisc .1\. onto a domain Dc e 3 ; the image domain D provides the desired example. Note that whenever (1 + e)-I < Izl < (1 + e), then (z, Z-I, 0) E .1\. and consequently F(z, z-t, 0) = (z, 1,0) E D, but that not all the points ofthe one-dimensional disc.1\* = {(z, 1,0): Izl < 1 + e} are contained in D; in particular, it is easy to see that (0, 1,0) ¢ D. Now the restriction to .1\* of any polynomial f is necessarily holomorphic in that disc. Hence, by the maximum modulus theorem If(zo, 1, 0)1 ~ sup If(z, 1, 0)1

(3)

Izl=1

whenever (1 + et l < IZol < 1. The point (zo, 1,0) is contained in D, and the inequality (3) consequently holds for any f E f!lJD. On the other hand, not all of the disc .1\* is contained in D, so that the maximum modulus theorem cannot be applied to a general function f E (!)D. Indeed, the functions Zl and Z2 are hoi om orphic in .1\., and under the biholomorphic mapping F they are transformed into functions fl = Zl 0 F- I and f2 = Z2 0 F- I in (!)D. But fll.1\* = z while f21.1\* = Z-I, so that at least one of these functions actually does not satisfy the inequality (3) and hence is not contained in f!lJD. In this example the point A = (0, 1,0) ¢ D but the inequality If(A)1 ~ sUPZeD 1f(Z) 1 holds for all f E f!lJD. That is really the key to the failure of polynomial approximation, in the sense that f!lJD = (!)D ifno such equality holds. That can be made precise as follows.

8. DEFINITION.

An open subset D KeD the set

K = {A E en: Ip(A)1 is contained in D. The set

£;

en is polynomially convex

if for

every compact subset

~ IlpliK for all polynomials p}

K is called the polynomially convex hull of K.

The terminology is motivated by the observation that if linear functions are used in place of polynomials in the preceding definition, then the hull K is just the ordinary convex hull of K, so linear convexity is just the ordinary notion of convexity.

9. DEFINITION.

The polynomial polyhedron in radius tJ > 0 is the open subset P(PI' ... , Pr; tJ)

e

defined by r polynomials PI' ... ' Pr and of

= {Z E en: IZil < tJ, 1Pj(Z) 1 < tJ for 1 ~

i;;i; n, 1 ;;i; j;;i; r}

58

Volume I

Function Theory

It is clear that polynomial polyhedra are necessarily polynomially convex. These polyhedra are useful auxiliary entities in the investigation of general polynomially convex sets. As a preliminary for this investigation it is convenient to discuss some geometric properties of these polynomial polyhedra. Consider a fixed such polyhedron P(P1' ... , Pr; (5) s;; en, noting that if r ~ 1, P(P1' ... , Pr; 15) s;; P(P1' ... , Pr-1; 15) s;; en

In terms of the polynomial Pro introduce the complex analytic submanifold

en + 1 : Zn+1 -

M = {Z E

Pr(Zl"'" zn) = O} s;;

en +1

noting that this is indeed an analytic submanifold of dimension n in en + 1 . The natural projection E: en x e 1 --+ en when restricted to M describes a biholomorphic mapping M --+ en, the inverse of which is the mapping F: en --+ M defined by F(Z) = (Z, Pr(Z)), The intersection M n {P(P1' ... , Pr-1; 15) x C} s;; en + 1 is mapped biholomorphically to P(P1"'" Pr-1; (5) under the projection E, and the intersection M n {P(P1' ... , Pr-1; 15) x A(O; t5)} s;; e n + 1 is mapped biholomorphically to P(P1' ... , Pr; 15) under the projection E. Thus, the polynomial polyhedron P(P1' ... , Pr; 15) defined by r polynomials can be identified with the complex analytic submanifold M of the polynomial polyhedron P(P1' ... , Pr-1; 15) x L\(O; 15) defined by r - 1 polynomials. This is useful in induction arguments on r, noting that a polynomial polyhedron defined by zero polynomials is just a polydisc. The geometric situation can be kept in mind with the aid of Figure 6.

M

T

r---i---------I

I

I

I

I

I

·Jl)-+----r---~~--~-_-_-_--_-_-_-_-_-_-_+l-_-_-_-Jr:--~c. 1------ P(p\, .. ·,Pr;b)

-I

Figure 6

10. THEOREM.

If cp is a Coo differential form of_bidegree (p, q) defined on the polynomial polyhedron P(Pl' ... , Pr; 15) and such that ocp = 0 and q > 0, then for any e < 15 there is a Coo differential form 1/1 of bidegr!!e (p, q - 1) defined on the polynomial polyhedron P(Ph ... , Pr; 6) and such that cp = 01/1 there.

F Polynomial Approximation

59

Proof. The proof will be by induction on the number r of polynomials defining the polyhedron. When r = 0, the polynomial polyhedron is merely a polydisc, and the theorem reduces to Dolbeault's lemma, Theorem E3. For the induction step, consider a polyhedron P(PI' ... , Pr; c5) where r > O. Choose a e"" function jj in C" such that jj = 1 in an open neighborhood of P(PI' ... ' Pr; e) and jj = 0 in an open neighborhood of the complement of P(PI' ... , Pr; c5), and introduce the COO differential form ijJ in all of C" defined by ijJ(Z) =

{~(Z) 0, there exists a polynomial P such that Ip(Z)-

f(Z) I < e whenever Z E K. For this purpose it is enough merely to find a polynomial polyhedron P such that K c Po c D, where Po is the union of some of the connected components of P. Then the function fo in P defined by if Z if Z

E E

Po P - Po

is holomorphic in P, and the desired result follows immediately from Theorem 11. Since D is polynomially convex by hypothesis, K ~ D, and K is also compact since it is clearly closed and the coordinate functions in C" are bounded on K by their bounds on K. Choose an open set U in cn such that [] is compact and K ~ u ~ [] ~ D. For each point A on the boundary au there exists a polynomial PA such that IPA(A)I > 1 > SUPZeK IPA(Z)I, since Art. K. The open sets {Z E C": IPA(Z)I > I} cover the compact boundary au, so a finite number of these sets also serve to cover au. If PI' ... , p,. is the corresponding finite set of polynomials, it is clear that the polynomial polyhedron P(Pl' ... , p,.; 1) is disjoint from an open neighborhood of the boundary of U, and that K ~ P(Pl, ... , p,.; 1). The union Po of those connected components of P(Pl, ... , p,.; 1) contained in U is then the desired set, and that suffices to conclude the proof of the theorem. It should be pointed out that by using Theorem 10 and Corollary 12 in place of the corresponding results for polydiscs, the proof of Theorem E5 can be carried over to give a proof of the assertion that £p,q(D) = 0 for any polynomially convex subset D and any index q > O. Since even more general results will be derived later, though, it is not worth pursuing this observation any further here.

G Domains of Holomorphy and Holomorphic Convexity

One of the most interesting and characteristic phenomena observed in the study of holomorphic functions of several variables is the existence of pairs of open sets DeE ~ Cn such that every function that is holomorphic in D necessarily extends to a function that is holomorphic in the strictly larger set E; several examples of this have already been pointed out. It is clearly of some interest to determine those sets D for which no such extension is possible. As will later become apparent, such sets playa basic role in the function theory of several variables. 1. DEFINITION. A domain of holomorphy in Cn is an open subset D ~ en for which there exists at least one function f E (!)n that cannot be extended as a holomorphic function through any boundary point of D. Iff E (!)n, then f has a power series expansion about any point A ED, and that power series converges in any polydisc .1\(A; R) for which .1\(A; R) ~ D. The set D is a domain of holomorphy if there exists a function f E (!)n such that whenever the power series expansion of f about a point A ED converges in a polydisc .1\(A; R), then conversely .1\(A; R) ~ D. Note that if the power series expansion of a function f E (!)n about a point A ED converges in a polydisc .1\(A; R) not contained in D, this does not necessarily provide an extension of the function f to a function holomorphic in an open subset ofC" properly containing D, for D n .1\(A; R) may consist of several connected components and may be dense in .1\(A; R). Thus, the condition that D be a domain ofholomorphy is slightly stronger than the condition that there exists a function f E (!)n that cannot be extended to a holomorphic function in a properly larger domain. It is well known that the unit disc .1\(0; 1) c C 1 is a domain of holomorphy. The Weierstrass product formula can be used to construct a nontrivial holomorphic function f in .1\(0; 1) such that every boundary point of .1\(0; 1) is the limit of a sequence of zeros of f, or alternatively the explicit power series expansion f(z) = I.z·! is easily seen not to have any continuation beyond the unit circle. More generally, any open polydisc .1\(A; R) ~ C" is a domain of holomorphy, for if f is either of the functions of one variable just mentioned, then 01=1 f«zj - aj)/rj) is holomorphic in .1\(A; R) but cannot be continued beyond any boundary point. There is a useful general criterion for a set to be a domain of 61

62

Volume I

Function Theory

holomorphy, reminiscent of the condition of polynomial convexity introduced in the preceding section.

2. DEFINITION. An open subset D K ~ D the set

~

KD = {A ED: If(A)1 ~

en is holomorphically convex if for any compact subset IIfllK for all f

E

(DD}

is also compact. The set KD is the holomorphically convex hull of Kin D.

Note that the set KD is always bounded, since the coordinate functions in en are bounded on KD by their maximal values on K, and that the set KD is a relatively closed subset of D. Thus, the condition that KD be compact is really just that it be closed in en, or equivalently, that it be disjoint from an open neighborhood of aD. In studying holomorphically convex hulls, it is convenient to have a simple notation for the distance from a point inside D to the boundary of D.

3. DEFINITION.

For any open subset D

~

dD(Z) = sup{e E IR: B(Z; e)

en and any point ZED, let ~

D}

and (jD.R(Z) = sup{e

E

IR:

~(Z; R) ~

D}

where R = (rl' ... , rn ), rj > 0, is any fixed polyradius. For any compact subset K let

~

D,

dD(K) = inf dD(Z) ZEK

and

Thus, dD(Z) is the supremum of the radii of all balls centered at Z and contained in D, and hence is the usual Euclidean distance from Z to the boundary of D, and (jD,R(Z) is an analogous measure of the distance from Z to the boundary of D but defined by polydiscs having poly radii proportional to R rather than by balls. The distance function (jD,R is more useful in many contexts than is the usual Euclidean distance, since it reflects the analytic situation more closely. As a further simplification, the notation (jD will sometimes be used in place of (jD,I' where 1= (1, ... , 1). Both dD and OD,R are clearly continuous functions in the domain D, so the expressions dD(K) and OD,R(K) have well-defined finite values. Note that dD(K) is the distance from the subset K to the boundary of D in the usual sense, while

G Domains of Holomorphy and Holomorphic Convexity

63

bv,R(K) is the distance measured by polydiscs having polyradii proportional to R. Note also that if Kl ~ K 2, then dv(Kt> ~ dv (K2) and bv,R(K 1 ) ~ bv ,R(K 2).

4. LEMMA. If K is a compact subset of an open set D ~ en and if 15 = bv,R(K) for some polyradius R, then any function f E &v extends to a holomorphic function in ~(A; bR) for any point A E iv.

UA

Proof. Whenever 0 < 6 < 15, the set K. = eK K(A; 6R) is clearly a compact subset of D. Iff E &v, then it follows from Cauchy's inequalities, Theorem A5, that

for all multi-indices I whenever A E K. Since the partial derivatives of fare holomorphic in D, the same inequalities also hold for all points A E iv, by definition of the holomorphically convex hull. That clearly implies that the Taylor expansion of the functionf at any point A E iv must converge in the polydisc ~(A; 6R), and since that holds for all 6 < 15, the proof is thereby concluded.

5. THEOREM. An open subset D morphically convex.

~

en is a domain of holomorphy if and only if it is holo-

Proof. First suppose that D is holomorphically convex. Choose a sequence of compact subsets Kv c D such that Kv c interior K v+1 and Uv Kv = D; also choose a countable dense sequence of points Av E D, and for each point A v, let ~(Av; Rv) be a maximal polydisc centered at Av and contained in D. For each index v there exists a point Zv E ~(Av; Rv) such that Zv ¢ iv, since D is holomorphically convex and hence iv is compact. Since Zv ¢ iv, there exists a holomorphic function J. E &v such that IJ.(Z,) I > II fv 11K . After multiplying this function by a constant and then raising it to a sufficiently large power, it can even be assumed that J.(Zv) = 1 and that Ifv(Z)1 ~ v- 12- v whenever Z E Kv' The series Lv vfv(Z) is thus absolutely and uniformly convergent on any compact subset of D. Consequently the infinite product f(Z) = flv[l - fv(Z»)" converges to a nontrivial holomorphic functionf E &v, and by construction this function is of total order at least v at the point ZV' It will be demonstrated that this function f cannot be extended as a holomorphic function across any boundary point of D, and hence that D is a domain of holomorphy. Suppose contrariwise that for some point A E D the function f extends to a holomorphic function in a polydisc ~(A; R) not contained in D. There is a subsequence ZVj such that the points ZVj lie in the connected component of D (\ ~(A; R) containing A and converge to a point Zo E aD (\ ~(A; R). The function f is of total order at least Vj at the point Zv.; hence, for any multi-index I, J

since this partial derivative is zero at ZVj whenever

Vj>

In That means that the

64

Volume I

Function Theory

function f is identically zero, which is a contradiction; hence, every holomorphically convex domain is a domain of holomorphy. Next suppose that D is a domain ofholomorphy, and choose a function f E (9D that cannot be extended as a hoi om orphic function across any boundary point of D. If K £; D is a compact subset of D and (j = (jD R(K), it follows from Lemma 4 that A(A; (jR) £; D whenever A E K D; for f is holom'orphic in A(A; (jR) by that lemma, but f cannot be extended as a holomorphic function beyond any boundary point of D. Thus KD must be disjoint from an open neighborhood of aD; that suffices to show that D is holomorphically convex and therefore to conclude the proof of the theorem. Actually the second part of the proof of the preceding theorem demonstrated a somewhat more precise assertion than merely that a domain of holomorphy is holomorphically convex. If D £; en is a domain of holomorphy, then for any compact subset K and any polyradius R,

6. COROLLARY.

£;

D

(1)

and

(2) Proof. Since K £; K D, the inequalities dD(K) ~ dD(K D) and (jD,R(K) ~ (jD,R(K D) are trivial. On the other hand, in the second part of the proof of Theorem 5 it was demonstrated that A(A; (jD R(K)· R) £; D whenever A E K D, so that (jD R(K) ~ (jD,R(K D) and hence (jD,R(K) = bD,R(K D), demonstrating (2). To derive the ~quality (1) from this one, note that

dD(Z)=sup{eEIR:Z+eWEDforall WE Cn with IIWII ~ I} =

inf dD , w(Z) IIWII=l

where dD,w(Z) = sup{e E IR: Z + rW E D for all r E IR with Irl ~ e}

Hence it is sufficient to show that inf dD,w(Z) = in[ dD,w(Z) ZeK

ZeK D

for any fixed W E c n with II W II = 1. After a nonsingular linear change of coordinates in en it can be assumed that W = (1, 0, ... ,0). The line segment from Z - W to Z + W is then just the intersection of the monotonically decreasing sequence of

G Domains of Holomorphy and Holomorphic Convexity

65

polydiscs .1(Z; RJ, where R. = (1 + (1/v), 1/v, ... , 1/v), and consequently it is clear that dD.w(Z) = lim. bD.R,(Z). Actually since .1(Z; R.) ;2 .1(Z; R.+ 1 ), the functions bD • R are a monotonically increasing sequence of continuous functions in D, and hence these functions converge uniformly on any compact subset of D. From this observation and the already demonstrated equality (2) it follows that inf dD • w(Z) = lim lim bD,R'(Z)

ZEK

•

=

ZEK

lim inf bD R (Z) = inf dD w(Z) v

ZeK D

."

ZeK D

'

and that suffices to conclude the proof. Theorem 5 is perhaps the most useful criterion that an open subset D ~ a domain of holomorphy and easily leads to still other criteria as follows.

en be

7. THEOREM. An open subset D ~ en is a domain of holomorphy if and only if for any discrete sequence of distinct points A. E D there exists a function f E (!)D such that lim sup.lf(A.)1 = 00. Proof. First suppose that D is a domain of holomorphy and consider a discrete sequence of distinct point A. E D. Choose a sequence of compact subsets K. c D such that K. c interior K.+1 and U. K. = D. Since D is holomorphically convex by Theorem 5, the holomorphically convex hulls K•. D are also compact subsets of D and hence cannot contain the entire sequence {A.}; so by passing to subsequences of the sequences of points {A.} and subsets {K.} it can be assumed that A. rf= K. D but A. E K.+ 1 • D • From the definition ofthe holomorphically convex hull of a set,'it follows that for each v there exists a function f. E (!)D such that If.(A.)1 > II f.IIK' After multiplying this function by a constant and then raising it to a sufficiently large power, it can even be assumed that (3)

and .-1

1f.(A.)1 > v +

L

,,=1

IfiA.)1

(4)

Whenever v ~ J-l it follows from (3) that 11f.IIK. ~ Ilf.k < 2-'; hence, the series f = L. f. converges in the topology of (!)D to a function f E (!)D' For this function, (5)

If J-l ~ v + 1, note that. A. E

K.+ 1 • D~ K".D; hence, by (3) necessarily If,,(A.)1 ;;:;; 1I.t;.IIK. < 2-", and by thiS and (4), it follows from (5) that

66

Volume I Function Theory

and hence that lim vIf(AJI = 00. Next suppose that D has the property that for any discrete sequence of distinct points Av E D there exists a function f E (YD such that lim supv If(Av)1 = 00. If Dis not a domain of holomorphy and hence by Theorem 5 is not holomorphically convex, there must be a compact subset KeD such that KD is not compact. For any discrete sequence of distinct points Av E K D' there is a function f E (YD such that lim sup vIf(Av)1 = 00; but then If(AJI > IlfilK for some values of v, contradicting the assumption that Av E KD • Thus, D must be a domain of holomorphy, and that concludes the proof. It should perhaps be noted that if D is a domain of holomorphy and {Av} is a discrete sequence of distinct points of D, there actually exists a function f E (YD such that limy If(Av)1 = 00, a rather stronger assertion than lim sup vIf(Av)1 = 00. This stronger assertion is not that much more useful, though, and is a special case of a more general result which will be demonstrated later; so this point will not be pursued further here.

8. THEOREM. If an open subset D ~ en is not a domain of holomorphy, there exist a point A ED and a polydisc L\(A; r) not contained in D such that for every function f E (YD the power series expansion off about the point A converges throughout L\(A; r). Proof. If D is not a domain of holomorphy, then by Theorem 5 it is not holomorphically convex, so there exists a compact subset KeD such that KD is not compact and hence such that bD(K D) = O. For a point A E KD such that bD(A) < bD(K) = r, the polydisc L\(A; r) is not contained in D. It follows from Lemma 4 that for every function f E (YD the power series expansion of f about the point A converges throughout L\(A; r), which was to be proved. Although the preceding theorem is a simple consequence of already established results, it is nonetheless worth stating separately. On the one hand, it shows that if an open subset D ~ en is not a domain of holomorphy, then all functions f E (YD can simultaneously be extended to a properly larger manifold; but, as will be observed in the next section, this larger manifold may not be an open subset of en. On the other hand, by the same light if for each point A E iJD there is at least one function fA E (YD that cannot be extended through A, then D is necessarily a domain of holomorphy. This provides an apparently simpler characterization of domains of holomorphy, one that is sometimes used as the definition of domains of holomorphy. Examples. The criterion provided by Theorem 5 can be used to show that a number of standard domains D ~ en are domains of holomorphy, as follows.

9. THEOREM.

Any open subset D ~

e 1 is a domain of holomorphy.

Proof. It is sufficient. to show that any open subset D ~ e 1 is holomorphically convex. For any compact subset K ~ D and any point a E iJD, the function (z - a)-l

G Domains of Holomorphy and Holomorphic Convexity

67

is holomorphic in D and bounded on K, but in modulus tends uniformly to 00 as Z approaches a; hence, Kn is disjoint from an open neighborhood of aD and is consequently compact. That shows that D is holomorphically convex as desired.

10. THEOREM.

Any open subset D domain of holomorphy.

£;

en

that is convex in the usual geometric sense is a

Proof. If D is an open convex set in en, then for any compact subset K £; D and any point A E aD there is a real-valued linear function tA(Z) = tA(X, Y) of the real coordinates in en = 1R 2 n such that tA(A) = 0 and tA(Z) < 0 whenever Z E K. When expressed in terms of Zj and Zj, this linear function can be written in the form tA(Z) = C + t Lj(CjZj + cjz) for some complex constants c, Cj, and cj. The condition that tA(Z) be real for all Z implies that C = c and cj = Cj; hence, tA(Z) = Re[c + LjCjzJ is the real part of the complex linear function C + LjCjZj' Now £(Z) = exp(c + LjCjz) is holomorphic in D, and fA(A) = 1 while IfA(Z) I = exp(tA(Z» < 1 for all Z E K and hence for all Z E Kn. Thus, again Kn is disjoint from an open neighborhood of aD, so Kn is compact and D is holomorphically convex. That suffices to conclude the proof.

11. THEOREM.

A complete Reinhardt domain having a logarithmically convex base is a domain of holomorphy.

Proof. Let D be a complete Reinhardt domain having a logarithmically convex base B, and let K be a compact subset of D, which can always be assumed to contain A(O; R.) such that the origin. There is a finite union of open polydiscs U' = K £; U' £; D; the set U' is also a complete Reinhardt domain, and the logarithmically convex hull Uo of its base U~ is the base of a complete Reinhardt domain U such that K £; U £; D. This domain U has the property that there exists a constant e > 0 such that whenever A = (aI' ... , an) E au and aj = 0 for some index j, then Ax = (aI' ... , Xj' .•• , an) E au whenever 0 ~ Xj < e. By iterating this observation, it follows that to every point A E au there can be associated a family of points Ax E au for some parameter values 0 ~ xi!' ... , x jk < e, such that Ax = A when xi! = ... = x jk = 0 and no coordinate of Ax is zero when Xj, ... Xjk > O. The point 10glAxi =(loglall, ... ,loglxjl, ... ) then lies on the boundary of the convex set log Uo = {(log Xl"'" log Xn): X E Uo and Xl'" Xn =F O}. The projection KB = {IZI=(lzII, ... ,lznl):ZEK} of the compact subset KeD to the base B is a compact subset KB £; B, and log KB = {loglZI = (lOgIZII, ... , loglznl): Z E K and Zl ... Zn =F O} is a closed set in the interior ofthe convex set log Uo. For any point A E au there is a real linear function tA(S) = v + LjVjSj such that tA(loglAxl) = 0 and tA(S) < 0 whenever S E log K B • Note that it can be assumed that tA is independent of the variable Sj whenever aj = O. Since the coefficients of the linear function tA can be varied slightly so that VI' ... , Vn become rational, and tA can then be multiplied by any positive integer, it can also be assumed that VI' ... , Vj are integers. Since K contains the origin and tA(S) < 0 for all S E log K B , all the integers Vl , ... , Vj are necessarily nonnegative. The function fA(Z) = e z~"" is then holomorphic in e" and hence in D, and I£(Z)I = exp tA(S) where Sj = loglzjl. Therefore, I£(A)I = 1 and IfA(Z) 1 < 1 whenever Z E K and Zl ... z" =F 0, and the same

Uv

V •

z:n

68

Volume I

Function Theory

inequality holds for all points Z E KD by definition of the holomorphically convex hull K D • That means that KD is disjoint from an open neighborhood of au and hence that KD is compact. Therefore, D is holomorphically convex, which suffices to conclude the proof. As demonstrated in Theorem B2, the domain of convergence of any power series expansion about the origin in en is a complete Reinhardt domain having a logarithmically convex base. The last result shows that for each such domain there is a function that is holomorphic in that domain but not in any properly larger domain and hence that there is a power series having that domain as precisely its domain of convergence. That completes the description of the domains of convergence of power series in several complex variables.

12. THEOREM.

The intersection of any two domains of holomorphy is also a domain of holomorphy.

Proof. If Dl and D2 are domains of holomorphy and K s;;; Dl n D2 is a compact subset, then KDlnD2~S;;; KDI n K D2 , since there are more holomorphic functions on a smaller set. Thus, KD I nD 2 is disjoint from an open neighborhood of aD l U aD2 and hence KDlnD2 is compact and Dl n D2 is holomorphically convex. That suffices to complete the proof. ~

13. DEFINITION. An open subset D s;;; en is locally a domain of holomorphy if each point A E aD has an open neighborhood U such that Un D is a domain of holomorphy. If D is a domain ofholomorphy and A E aD, then any ball B(A; 6) is a domain ofholomorphy by Theorem 10, and hence any intersection B(A; 6) n D is a domain ofholomorphy by Theorem 12; thus, any domain of holomorphy is locally a domain of holomorphy. It would clearly be very useful if the converse were also true, since that would show that the property of being a domain of holomorphy is a purely local property of the boundary of a domain. Actually the converse assertion is true, but that is a rather deeper result and will not be proved until some further machinery has been developed. The problem of whether an open subset of en that is locally a domain of holomorphy is actually a domain of holomorphy is often called the Levi problem in en, since it first arose in about 1910 in connection with an investigation by E. E. Levi oflocal conditions for a subset of en having a smooth boundary to be a domain of holomorphy. The study of this problem and of its natural generalizations has been one of the principal themes in the development of the theory of hoI om orphic functions of several variables. There are still other criteria for an open subset of en to be a domain of holomorphy, one of which involves the Dolbeault cohomology groups introduced in section E. This criterion provides another example, to add to those discussed in the preceding two sections, of the usefulness of the operator in a variety of problems in complex analysis.

a

G Domains of Holomorphy and Holomorphic Convexity

14. THEOREM. If D is an open subset of en for which Ye°·q(D) = D is a domain of holomorphy.

°for 1

~ q ~ n-

69

1, then

Proof. The proof will be by induction on the complex dimension n. When n = 1 the hypothesis involving the Dolbeault cohomology groups Ye°,q(D) is vacuous and hence is automatically satisfied for any open subset D s; e 1. But by Theorem 9, any open subset D s; e 1 is a domain ofholomorphy, so the present theorem holds in this case. Assume that the theorem has been proved to hold for open subsets of en- 1 for n ;?; 2, and consider an open subset D s; en for which Ye°·q(D) = for 1 ~ q ~ n - 1. As a useful preliminary observation it will first be demonstrated that for any complex linear submanifold L s; en of dimension n - 1, each connected component of the intersection D n L, viewed as an open subset of L = en-I, is a domain of holomorphy. In view of the inductive hypothesis, it is enough for this purpose just to show that Ye°·q(D n L) = for 1 ~ q ~ n - 2. So consider a a-closed coo differential form t/J of bidegree (0, q) on D n L, where 1 ~ q ~ n - 2. To simplify notation perform a translation and a nonsingular linear change of coordinates in en so that L = {Z E en: Zn = o}. It is clear that the differential form t/J can be extended to a a-closed coo differential form of bidegree (0, q) in an open neighborhood U of the closed subset D n L s; D merely by viewing t/J as a differential form independent of the variable Zn' For a Coo function p in D, such that p is identically equal to one in an open neighborhood of D n L in D and that the support of p is contained in U, introduce the differential form is a a-closed Coo differential form in all of D, since ap is identically equal to zero in open neighborhoods of D n iJU and of D n L = {Z ED: zn = O}. Since Ye 0 .q+l(D) = 0, it follows that there is a Coo differential form 'iI of bidegree (0, q) in D such that r/Jl, and BM(Aj- l ; r/2Jl) n BM(Aj; r/2Jl) #- ,p for j = 1, ... , v}. The sets D/J,. are clearly open in M. Furthermore, since M is connected, necessarily M = U/J,.D/J,., for if A~ E M is not contained in this union, then the same construction based at A~ yields an open subset of M containing A~ and disjoint from this union, so the complement is also open. In order to complete the proof it is enough to show that D/J,. is compact, since that implies that D/J,. is second c~untable. Since D/J,l s;; BM(Ao; r/Jl) and r/u < dM(A o), the set D/J,l is compact. If D/J,. is compact, then choose finitely many points Zj E D/J,. such that D/J .• ~ UjBM(Zj; r/2Jl). Then clearly D/J,.+l ~ D/J,. u UiB(Zj; r/Jl), so that

74

Volume I

Function Theory

15,..• +1 is compact. That shows by induction on v that all the sets 15,..• are compact and concludes the proof. Since a Riemann domain M is second countable, the space (!}M of all holomorphic functions on M is a Frechet space with the natural topology as described in Definition Fl. The space (!}M is actually quite large, for the projection mapping P: M -+ (;n induces an injective homomorphism P*: (!}cn -+ (!}M' However, the functions in (!}M need not separate points, in the sense that there may be distinct points Al and A2 of M such that f(Ad = f(A 2) for all f E (!}M; but of course in this case necessarily P(Ad = P(A 2). To see a simple example, consider the domains Dl = {Z E (;2: 1 < IIZII < 3} and D2 = {Z E (;2: -2 < Xl < 2, Ihl < e, IZ21 < e}, noting that Dl n D2 consists of two connected components D~2' D~2' and that D2 is contained in the convex hull of Dl whenever e is sufficiently small. Let D be the Riemann domain consisting of the disjoint union of Dl and D2 , with points of Dl and D2 identified if they are contained in D~ 2 but not if they are contained in D~2' Thus, D is a Riemann domain, where P-l(Z) consists of two points whenever Z E D~2 but otherwise of one point, as indicated schematically in Figure 8. Every function f E (!}D viewed as a function in (!}D, extends to a unique holomorphic function in the convex hull of D 1 , and hence must have the same value at the two points P-l(Z) whenever Z E D~2'

t

Figure 8

4. THEOREM. For any Riemann domain M with projection P there exist a Riemann domain Ml with projection P1 and a nonsingular holomorphic mapping F: M -+ Ml such that P1 0 F = P, that F induces an isomorphism F*: (!}M, -+ (!}M, and that functions in (!}M, separate points in M 1 • Proof. Introduce an equivalence relation on M by setting A ~ B whenever f(A) = f(B) for all functions f E (!)M' noting that this is indeed an equivalence relation in the usual sense. Let Ml = M/~ be the quotient space of M under this equivalence relation, and F: M -+ M 1 be the natural mapping that associates to any point A E M its equivalence class. It is evident that A ~ B implies that P(A) = P(B); hence, P

H Envelopes of Holomorphy and Riemann Domains

75

induces a natural mapping Pl : Ml -+ en such that P = Pl 0 F. In order to complete the proof ofthe theorem, it suffices to show that whenever A '" Band BM(A; e) S; M, BM(B; e) S; M, then A' '" B', for A' E BM(A; e), B' E BM(B; e) if and only if P(A') = P(B'). Now whenever f E (9M' then al11j"IazI E (9M for any multi-index I, where differentiation is with respect to the natural coordinates imposed on M by the projection mapping P. If A '" B, then

for any multi-index I. Thus, the functions f 0 (PIBM(B; en-l and f 0 (PIBM(B; en- 1 have the same Taylor expansions about P(A) = P(B), so that f(A') = f(B') whenever P(A') = P(B'). That suffices to conclude the proof. As a consequence of this observation, when the primary interest is in the holomorphic functions on a Riemann domain, there is no loss of generality in restricting attention to those Riemann domains for which holomorphic functions separate points, and that is frequently done. Next for the problem of the extension of hoI om orphic functions, it is convenient to introduce the following definition.

5. DEFINITION. A complex manifold E is a holomorphic extension of a complex manifold M if M is an open subset of E with the induced complex structure and the natural restriction mapping r: (9E -+ (9M is an isomorphism. If E is a holomorphic extension of M, then each connected component Ev of E must contain a connected component Mv of M, since otherwise the restriction mapping r cannot be an isomorphism, and Ev is clearly a holomorphic extension of Mv. Thus, in discussing holomorphic extensions, it is sufficient to consider only connected complex manifolds. If E is any connected complex manifold containing M as an open subset, then the restriction mapping r is necessarily injective, by the identity theorem. Hence, E is a holomorphic extension of M precisely when the restriction mapping r is surjective, or equivalently, precisely when every holomorphic function on M extends to a holomorphic function on E. As noted earlier, an open subset of en may have a Riemann domain as a holomorphic extension, but holomorphic extensions of Riemann domains are necessarily Riemann domains, as is clear from the following observation.

6. THEOREM. If M is a Riemann domain with projection P and E is a holomorphic extension of M, then E is also a Riemann domain, with a projection p* such that p* IM = P. Proof. If M is a Riemann domain with projection mapping P: M -+ en, then the coordinate functions of the mapping P extend to holomorphic functions on E, so the mapping P extends to a holomorphic mapping P*: E -+ e. The determinant of the Jacobian of this mapping p* is a holomorphic function p* = det Jp • on E, and. it is nowhere zero on M. That implies that p* is also nowhere zero on E and hence that P*: E -+ en is nonsingular, for if p*(A) = 0 for some point A E E - M, then IIp* is holomorphic on M but does not extend to a holomorphic function on E,

76

Volume I

Function Theory

contradicting the assumption that E is a holomorphic extension of M. That suffices to conclude the proof. If holomorphic functions separate points of a Riemann domain M, that is not necessarily true for all holomorphic extensions of M. The example of Figure 8 illustrates this, since the Riemann domain D is a holomorphic extension of the open subset Dl c 1C 2 • However, in any such case the reduction described in Theorem 4 can be applied to the holomorphic extension E without altering the subset M c E, yielding a hoi om orphic extension El for which the holomorphic functions do separate points. It is thus evident that it is quite reasonable to restrict attention to the category of connected Riemann domains for which the holomorphic functions do separate points; within this category there does always exist a maximal holomorphic extension. This maximal holomorphic extension can be constructed quite explicitly in terms of the Frechet algebra of holomorphic functions on the given Riemann domain. To carry out the construction, consider an arbitrary connected Riemann domain M with projection P: M -+ en, not necessarily assuming that holomorphic functions separate points of M. The set of all continuous algebra homomorphisms T: (9M -+ IC is called the spectrum of the Frechet algebra (9M and is denoted by spec (9M. As might be expected, there is a natural one-to-one correspondence between spec (9M and the set of closed maximal ideals in (9M by associating to each T E spec (9M the ideal in (9M that is the kernel of the homomorphism T, but this result is not needed here and so will not be proved until later. For each point A EM, the evaluation mapping SA: (9M -+ IC defined by SA(f) = f(A) is clearly an element SA E spec (9M. The correspondence that associates to any point A E M the evaluation mapping SA E spec (9M is the natural mapping S: M -+ spec (9M. If the holomorphic functions on M separate points, the natural mapping S: M -+ spec (9M is clearly injective and so can be viewed as imbedding M in spec (9M. If E is a holomorphic extension of M, then the restriction mapping rE : (9E -+ (9M must actually be an isomorphism of Frechet algebras, for it is obviously a continuous mapping between Frechet spaces and is an algebraic isomorphism by definition of holomorphic extension and hence by the open mapping theorem must be an isomorphism of Frechet spaces. The inverse mapping ri 1 : (9M -+ (9E is thus also an isomorphism of Frechet algebras. For each point A E E the mapping S;: (9M -+ IC defined by S;(f) = (ri 1f)(A) is then clearly an element S; E spec (9M' and the correspondence that associates to any point A E E this composition S; E spec (9M is a well-defined mapping SE: E -+ spec (9M. If the holomorphic functions on E separate points, which can of course happen only if the holomorphic functions on M already separate points, the mapping SE: E -+ spec (9M is also injective, so can be viewed as imbedding E in spec (9M. In that case the imbedding S: M -+ spec (9M is merely the restriction S = SEIM. Next to any element f E (9M there can naturally be associated a function J on spec (9M by defining J(T) = T(f) for any T E spec (9M. In particular, to the coordinate functions PI' ... , Pn of the projection P: M -+ en there can be associated the functions PI' ... , Pn on spec (9M' and these functions can in turn be taken as the coordinate functions defining a mapping P: spec (9M -+ en. Note that as a consequence ofthese definitions the composition ofthe natural mapping S: M -+ spec (9M

H Envelopes of Holomorphy and Riemann Domains

77

en

with this mapping P: spec (!)M -+ is just the original projection P = po S. Note further that for any hoi om orphic extension E of M, the projection P* ofthe Riemann domain E must be given by P* = ri 1 (p). Hence, the composition of the mapping SE: E -+ spec (!)M with the mapping P: spec (!)M -+ is just the projection P* = poSE.

en

en

7. LEMMA. For any connected Riemann domain M with projection mapping P: M -+ the set spec (!)M can be given the structure of a Riemann domain for which the projection mapping is P: spec (!)M -+ en, and holomorphic functions on this Riemann domain separate points. Proof. Note first that for any fixed point T E spec (!)M there is a compact subset K of M such that IT(f)1 ~ IlfilK for all f E (!)M' Indeed if that were not so, then, when M = U. K. for some compact subsets K. £ M with K. contained in the interior of K.+ 1 , there would exist for each index v afunctionf. such that IT(f.) I > IIf.IIK,. After multiplying f. by a suitable constant and raising it to a sufficiently high power, it could even be assumed that T(f.) = 1 and Ilf.IIK, < T'. The series L.J. then converges in the topology of (!)M, but the series L. T(f.) diverges, contradicting the continuity of T. Now choose a positive number e such that e < t5M ,R(K) for the polyradius R = (1, ... , 1), and note that K. = UZEKL\(Z; eR) is a compact subset of M. It follows from the Cauchy inequalities, Theorem A5, that

for any function f E (!)M, any point A E K, and any multi-index I, where as usual differentiation is with respect to the natural coordinates imposed on M by the projection P. Consequently,

for any f

E (!)M'

The power series (1)

therefore converges for all points Z estimates,

E

A(P(T); eR) for any

f

E (!)M'

By the usual

so that for any fixed point Z E A(P(T); eR) the mappingf -+ LT(f, Z) is a continuous linear mapping LT(Z): (l)M -+ C. By Leibniz's rule for differentiation, the derivatives iJllIj/iJZ1 behave on a product function in the same way as the corresponding terms

78

Volume I

Function Theory

in the power series expansion of a product; since T is an algebra homomorphism, then LT(fg, Z) = LT(J, Z)LT(g, Z), and consequently LT E spec (!)M' The correspondence Z -+ Lr(Z) thus defines a mapping L T: .1.(P(T); eR) -+ spec (!)M' It is clear from (1) that LT(P(T» = T. On the other hand, since if I = (0, ... , 0) if I = (0, ... , 1, ... ,0) with 1 in the jth place otherwise

OIIlpi _ {Pi OZI 1

o

it also follows readily from (1) that

and hence that P(LT(Z» = Z whenever Z E .1.(P(T); eR). The desired topology can then be introduced by taking as the basic open neighborhoods in spec (!)M the sets L T(.1.(P(T»; eR) for all points TE spec (!)M and all sufficiently small positive numbers e. It is of course necessary to show that these sets satisfy the appropriate conditions to define a topology. In this case it is clearly sufficient merely to show that if T' = LT(Z') for some point Z' E .1.(P(T); eR), then LT(Z) = LT'(Z) whenever Z is sufficiently near Z'. For any function f E (!)M' note from (1) that

Hence the Taylor expansion of the function LT(J, Z) about Z' has the form

=

6)! :! T(:~:!)(Z

- P(T'»J(Z' - P(TW

since Z' = p(T'). On the other hand, T'(f) = LT(f, Z'), so from (1) again LT·(f, Z)

=

1 (OIJ'J) L.., T' ~J (Z J. uZ J

~

P(T'W

~ = ~ J!1 LT (OIJ'J) oZJ' Z' (Z - P(T'W

=

, h J!1 I!1 T (OII+J'J) azI+J (Z -

~

I

~,

P(T» (Z - P(T

» J

H Envelopes of Holomorphy and Riemann Domains

79

Thus, LT(f, Z) = LT'(f, Z) for all f E (!)M and all points Z sufficiently near Z', so = LT'(Z) for all points Z sufficiently near Z' as desired. With the topology thus introduced, it is evident that spec (!)M has the structure of a Riemann domain with projection F: spec (!)M -+ en; for LT is a homeomorphism from an open neighborhood of F(T) in en to an open neighborhood of T in spec (!)M and is the local inverse of the mapping F. It only remains to show that holomorphic functions separate points on this Riemann domain. For any f E (!)M the induced functionjis hoI om orphic on spec (!)M' since whenever T E spec (!)M' the compositionj(LT(Z)) = LT(f, Z) is a holomorphic function of Z near F(T); so by the definition of the spectrum of the Frechet algebra (!)M these functions j separate points on spec (!)M. That suffices to conclude the proof. LT(Z)

8. THEOREM.

In the category of Riemann domains for which holomorphic functions separate points, any such Riemann domain has a unique maximal holomorphic extension.

Proof. Let M be a Riemann domain with projection P: M -+ en, and assume that holomorphic functions separate points on M. It follows from Lemma 7 that spec (!)M has the structure of a Riemann domain with projection F: spec (!)M -+ en, and that holomorphic functions separate points on spec (!)M as well. It was noted before that the natural mapping S: M -+ spec (!)M is injective in this case. Since it was also noted that F 0 S = P, and Fand P are both nonsingular holomorphic mappings, it follows that S is a nonsingular holomorphic mapping as well. Thus, S is a biholomorphic mapping between M and its image S(M) ~ spec (!)M and can be viewed as imbedding M as an open subset of spec (!)M with the induced complex structure. Let E(M) be the union of those connected components of the complex manifold spec (!)M that meet the subset S(M) ~ spec (!)M; then S(M) is also an open subset of the complex manifold E(M) with the induced complex structure, and the natural restriction mapping (!)E(M) -+ (!)S(M) is injective. It was noted in the proof of Lemma 7 that for any function f E (!)M the induced functionjis holomorphic on spec (!)M. Sincejo S = f, it follows that the restriction mapping (!)E(M) -+ (!)S(M) is surjective as well and hence that E(M) is a holomorphic extension of S(M) = M. If E is any other holomorphic extension of M, where E is a Riemann domain with projection P*: E -+ en and holomorphic functions also separate points on E, then as noted before the mapping SE is also injective; so since F 0 SE = P*, it follows that SE is a biholomorphic mapping between E and its image SE(E) ~ spec (!)M. Since S = SEIM, it follows that S(M) ~ SE(E), and since each connected component of SE(E) must contain a connected component of M, it is furthermore clear that SE(E) ~ E(M) ~ spec (!)M. That shows that E(M) is a maximal holomorphic extension of M in this category, and since the uniqueness is obvious, that suffices to conclude the proof.

9. DEFINITION.

If M is a Riemann domain for which holomorphic functions separate points, the envelope of holomorphy E(M) of M is the maximal holomorphic extension of M among all those holomorphic extensions for which holomorphic functions separate points.

The existence of the envelope ofholomorphy and its uniqueness up to biholomorphic mappings are established in Theorem 8. That shows in particular that to

80

Volume I

Function Theory

every open subset D s en there corresponds a unique maximal holomorphic extension E(D) of D within the category of Riemann domains for which holomorphic functions separate points, but as shown by the example of Figure 7, E(D) need not be representable as an open subset of en containing D. The example of Figure 8 also indicates that there can be no analogous theorem without the assumption that holomorphic functions separate points. Actually the proof of Theorem 8 demonstrated somewhat more than what the statement of the theorem indicated, since the envelope of holomorphy E(M) of the Riemann domain M was constructed quite explicitly in terms ofthe spectrum of the Frechet algebra (!)M. This additional information was not emphasized because it is still incomplete in two quite interesting points. First, the envelope of holomorphy E(M) was constructed as a subset of spec (!)M' leaving open the question whether E(M) = spec (!)M. It was shown that the evaluation mapping at each point A E E(M) represents a point in the spectrum spec (!)M' but the question remaining is whether conversely every homomorphism in spec (!)M is just the point evaluation at some point of E(M). That is the case, as will be demonstrated in Theorem HU1 and consequently actually E(M) = spec (!)M. Second, spec (!)M was given the structure of a Riemann domain with projection P: spec (!)M -+ en by imposing an appropriate topology on the set spec (!)M' as described in Lemma 7; but that leaves open the . question of how this topology compares with any more natural or intrinsic topologies on the set spec (!)M. Now the construction of spec (!)M leads immediately to the functions] on spec (!)M associated to the functions f E (!)M. These functions] are all hoI om orphic in the complex structure imposed on spec (!)M' and indeed they are precisely the holomorphic functions on spec (!)M' since it is a hoi om orphic extension of M. The most natural topology to introduce on spec (!)M is the weakest topology in which all these functions]are continuous. The question is then whether this is a weaker topology than that imposed on spec (!)M in Lemma 7. It will be demonstrated in Theorem HIR5 that these topologies actually coincide-that is, that the topology of the complex manifold spec (!)M can be described as the weakest topology in which all global holomorphic functions are continuous. Thus, the eventual result will be that for any Riemann domain M for which holomorphic functions separate points, the envelope of holomorphy E(M) can be identified naturally with the spectrum of the Frechet algebra (!)M' where spec (!)M is given the weakest topology in which all the functions] for f E (!)M are continuous.

I Riemann Domains of Holomorphy

To any open subset D £::: en there is canonically associated its envelope of holomorphy E(D), the maximal holomorphic extension of D in the category of Riemann domains for which holomorphic functions separate points. If E(D) is also an open subset of en, then it is clear from Theorem G8 that E(D) is a domain ofholomorphy. The envelopes of holomorphy of subsets in en, or more generally the envelopes of holomorphy of Riemann domains for which holomorphic functions separate points, are thus natural candidates to be considered the analogues for Riemann domains of domains of holomorphy. From this point of view it is fairly natural to limit the consideration to those Riemann domains for which holomorphic functions are assumed to separate points, as will be done whenever convenient in this section. The extent to which such an assumption is necessary will be discussed in a later section. As in the case of open subsets of en, there are a number of equivalent characterizations of domains of holomorphy among Riemann domains; the characterization chosen here as the definition is the one closest to Definition G 1 but is slightly more complicated to state. Recall that if M is a Riemann domain with projection P: M -+ en, then for any point A E M and polyradius R there are open neighborhoods L\M(A; sR) of A in M which are mapped biholomorphically by P to open polydiscs L\(P(A); sR) in en whenever s < M,R(A). For any holomorphic function f E (!JM' the composition fA = f 0 (PIL\M(A; sRW l is a well-defined holomorphic function in an open neighborhood of P(A) in en whenever s is sufficiently small, and the power series expansion of this function about the point P(A) is of course independent of s. This power series converges at least in the polydisc L\(P(A); sR) where s = M,R(A), and may for some functions f and some points A converge in a properly larger polydisc.

1. DEFINITION. The radius of convergence of the holomorphic function f A E M in terms of the polyradius R is

E

(!JM at the point

PM.R(f; A) = sup{s: the power series expansion of £. about P(A) converges in L\(P(A); sR)} 81

82

Volume I Function Theory

2. DEFINITION. A Riemann domain of hoiomorphy is a Riemann domain M for which there exists at least one function f E (9M such that PM.R(f; A) = c5M.R(A) for any polyradius R and any point A EM. Since it is always the case that PM.R(f; A) ~ c5M.R(A), the critical property in this definition is just that PM.R(f; A) ~ c5M. R(A); in a sense that amounts to the condition that f cannot be extended to a holomorphic function in any properly larger manifold. Obviously this definition reduces to Definition G 1 in the special case that M is an open subset of en. Any domain of holomorphy in en is thus an example of a Riemann domain of holomorphy. The following very simple observations are worth noting explicitly here.

3. LEMMA. A Riemann domain of holomorphy admits no properly larger holomorphic extension. Proof. Suppose to the contrary that M is a Riemann domain of holomorphy that does admit a properly larger holomorphic extension E. There must be some point A E Me E and polydisc ~E(A; BR) ~ E such that ~E(A; BR) is not contained in M and hence such that c5M.R(A) < B. Any f E (9M extends to a holomorphic function on E, so that evidently PM.R(f; A) ~ B > M.R(A). But since M is a Riemann domain of holomorphy, there exists at least one function f E (9M for which PM.R(f; A) = M.R(A), and that contradiction suffices to conclude the proof.

4. LEMMA. If for a Riemann domain M there exist a point A E M, a polyradius R, and a constant > M.R(A) such that PM,R(f; A) ~ for every f E (9M' then M admits a properly larger holomorphic extension. Proof.

Let E' be the disjoint union of the Riemann domain M and the polydisc R) ~ en. With the projection P of the Riemann domain M on the subset M ~ E' and the identity mapping on the polydisc ~(P(A); R) ~ E', the set E' has the structure of a Riemann domain. Any function f E (9M can be viewed as a holomorphic function on the subset M ~ E'. Since f 0 (PI~M(A; M.R(A)RW 1 is holomorphic in ~(P(A); M.R(A)R) and by hypothesis extends to a holomorphic function on ~(P(A); R), this provides an extension of f to a holomorphic function on all of E'. The class offunctions so constructed form a subalgebra (9;'. ofthe algebra (9E' of all holomorphic functions on E', and under the natural restriction mapping f E (9E' -+ flM E (9M' this subalgebra is isomorphic to (9M' Now introduce the equivalence relation defined by setting Z '" B where Z E ~(P(A); R) and B E M whenever f(Z) = f(B) for all f E (9;'.. It follows readily as in the proof of Theorem H4 that the space of equivalence classes E = E' / '" is a Riemann domain with (9E = (9;'., and it is evident that E is a properly larger extension of M. That suffices to conclude the proof. ~(P(A);

These two lemmas taken together suggest but do not quite demonstrate that Riemann domains of holomorphy are the same as Riemann domains admitting no properly larger holomorphic extension.s. For open subsets of en this equivalence

I

Riemann Domains of Holomorphy

83

was easily shown by using the auxiliary concept of holomorphic convexity; that notion is equally useful in the case of Riemann domains.

5. DEFINITION. A Riemann domain M is holomorphically convex if for any compact subset K S;;; M the set KM

=

{A EM: If(A)1 ~ IlfilK for allf E (l)M}

is also compact. The set KM is the holomorphically convex hull of Kin M.

The basic result that Riemann domains of holomorphy are the same as holomorphically convex Riemann domains is unfortunately considerably more difficult to prove than the corresponding result for open subsets of en. In one direction the proof is essentially the same, as follows.

6. THEOREM.

If M is a holomorphically convex Riemann domain, then M is a Riemann domain of holomorphy.

Proof. Suppose that M is a holomorphically convex Riemann domain, which can of course be supposed to be connected. Since M is second countable by Theorem H3, it is possible to choose a sequence of compact subsets K. s;;; M, such that K. S;;; interior K.+l and U. K. = M, and a countable dense sequence of points A. E M. The holomorphically convex hulls K. are compact, so if L\M(A.; R.) is a maximal polydisc in M centered at A., there must be a point Z. E L\M(A.; R.) such that Z. ¢ K•. Then, as in the first part of the proof of Theorem G5 it follows readily that there are functions f. E (l)M for which f.(Z.) = 1 and IIf.IIK. < v- 12-·. The infinite product f = 0.(1 - f.t is then a nontrivial holomorphic function on M and has total order at least v at the point Z •. In order to complete the proof, it is enough just to show that PM,R(f; A) ~ bM,R(A) for any polyradius R and any point A E M. Suppose to the contrary that PM,R(f; A) > bM,R(A) for some polyradius R and some point A EM. If Us;;; M is the connected component of P-l(L\(P(A); PM,R(f; A)R» containing L\M(A; bM,R(A)R), it is clear that L\{P(A); bM,R(A)R)

S;;;

P(U) c: L\(P(A); PM,R(f; A)R)

S;;;

en

with the second inclusion necessarily a proper inclusion, and that there exists a point BE L\(P(A); PM,R(f; A)R) such that BE oL\(P(A); bM,R(A)R) 11 oP(U). There must exist a subsequence {A •.J } of the countable dense sequence {A.} such that A •.J E U and P(A.) ~ B. Since B ¢ L\(P(A.); R •. ), the polyradii R •. must approach zero, so P(z.) ~ B and it can be assumed that L\~(A'j; R.) S;;; u. N~w the nontrivial function fA = f 0 (PI U)-l extends to a holomorphic function in all of L\(P(A); PM,R(f; A)R) and has total order ~ Vj at the points P(Z.) converging to B; but as in the proof of

Theorem G5 that is impossible, and with that contradiction the proof is concluded. The converse ofthe preceding theorem is considerably harder to demonstrate. The direct extension to Riemann domains of the second half of the proof of Theorem G5 really yields only the following weaker result.

84

Volume I

Function Theory

7. THEOREM. If M is a Riemann domain of holomorphy, then for any compact subset K eM and any polyradius R,

Proof. Since K ~ K M, it is of course trivial that -00 for almost all values r in the interval [rl' r2 ]. Note also that since F is harmonic, then M F.(a; r) = 0(. log r + P. for some real constants 0(., P•. Indeed, in terms of polar coordinates (r, 9) centered at a, the Laplacian can be written

Hence,

J

Subharmonic Functions

99

which implies that MFv does have the form asserted. The constants a v and Pv are determined by the two conditions M Fv(a; r) = fv(a; rj ) = (Lv log rj + Pv for j = 1, 2, and Mu(a; r) ~ (Lv log r + Pv whenever r 1 ~ r ~ r2 • The constants (Lv and Pv are monotonically decreasing as v tends to 00 and are bounded from below, since Mu(a; r) > - 00 for almost all r E [rl' r2 ], and hence converge to some finite values ex, P for which Mu(a; r) ~ ex log r + P whenever r 1 ~ r ~ r2 • Moreover, it follows from an application of Lebesgue's monotone convergence theorem that these limiting values are determined by the two conditions (L log rj + P= lim v (Lv log rj + Pv = lim v M!.«(L; rj ) = Mu(a; rj ) for j = 1, 2. That implies first that Mu(a; r) > - 00 and hence shows that u is integrable over every circle in D, and implies further that Mu(a; r) is convex as a function of log r. This completes the proof. The only property of subharmonic functions used in proving Theorems 3 and 6 was that the maximum theorem holds for the difference u - h between a subharmonic function u and a harmonic function h; the remainder of the arguments rested on familiar properties of harmonic functions. That suggests an alternative characterization of subharmonic functions, a characterization that can be expressed in a variety of ways, some of which are sometimes taken as the definition. For the purposes of this characterization, a mapping u: D -+ [ - 00, + 00) in a connected open subset D !;;; C will be said to satisfy the maximum principle in D if u(a) = sUPz E D u(z) for a point a E D only when u is constant in D. For an upper semicontinuous mapping u: D -+ [ -00, +00) in an open subset D s; C the following conditions are equivalent:

7. THEOREM.

(i) u is subharmonic in D. (ii) For any harmonic function h in a connected open subset U !;;; D, the difference u - h satisfies the maximum principle in U. (iii) Whenever h is a harmonic function in an open neighborhood of a compact subset K S; D and u(z) ~ h(z) at each point z E oK, then u(z) ~ h(z) at each point zEK. (iv) Whenev~ p is a complex polynomial and u(z) ~ Re p(z) at each point z E ol\(a; r) where l\(a; r) S; D, then u(z) ~ Re p(z) at each point z E l\(a; r). (v) Whenev~ p is a complex polynomial and u(z) ~ Re p(z) at each point z E ol\(a; r) where l\(a; r) S; D, then u(a) ~ Re p(a). Proof. It follows from Theorem 2(b) that condition (i) implies condition (ii), while it is quite obvious that condition (ii) implies condition (iii), which implies condition

100

Volume I

Function Theory

(iv), which in turn implies condition (v). Indeed, (iv) is a special case of (iii), since the real part of a polynomial is a harmonic function, while (v) is evidently a special case of (iv). It only remains to show that condition (v) implies condition (i). If u satisfies condition (v) for a disc /i(a; r) s; D, it suffices to verify the inequality (2) for that disc. Choose a monotonically decreasing sequence of continuous functions Iv on oLl(a; r) such that limv-+oo Iv (z) = u(z) at each point z E oLl(a; r), as in the proof of Theorem 3. As is well known, each function!. can be approximated uniformly on oLl(a; r) by trigonometric polynomials, merely summing the Fourier series expansion of Iv by arithmetic means and applying Fejer's theorem. Therefore, for each index v there is a trigonometric polynomial q., a finite sum of the form qv(O) = Ll avleil8, such that Ifv(a + re i8 ) - qv(O)1 < l/v for all real O. The ordinary polynomial Pv(z) = avo + 2Ll>o aVlr-l(z - a)l then has the property that Re pv(a + re i8 ) = qv(O); hence, IIv(z) - Re Pv(z)1 < l/v whenever z E oLl(a; r). Now since u(z) ~ Iv(z) < Re(pv(z) + l/v) whenever z E oLl(a; r), it follows from condition (v) that u(a) < Re(pv(a) + l/v). Using these various results and noting that (3) holds for the harmonic function Re Pv show that

-1 2n

1 2

0

"

u(a

+ re l8. ) dO =

1 lim -2 v

n

1 2

"

I.(a

+ re l8. ) dO

0

1Joe" [Re pv(a + re .l8 ) - ;IJ dO

~ limv inf 2n

~ limv inf [ Re pv(a) - ~J ~ limv inf [u(a) - ~J =

u(a)

Thus u is subharmonic, and that concludes the proof. Unlike harmonic functions, subharmonic functions are not necessarily smooth. However, for those subharmonic functions that are differentiable, there are still other parallels with harmonic functions, such as the following result. A real-valued function u of class C 2 in an open subset D s; C is subharmonic in D precisely when Llu = 4 02U/ OZ OZ ~ 0 throughout D.

8. THEOREM.

Proof. It is a minor calculation to verify that Ll = 02/0X 2 + 02/oy2 = 4 02U/ OZ oz. As a preliminary observation, note that if Llu > 0 throughout D, then u is subharmonic in D. Indeed, if u is not subharmonic, then by Theorem 7 there must exist a harmonic function h in D and a point a E D such that u - h has a local maximum value at a; but then the Hessian matrix of the function u - a must be negative semidefinite at the point a, an impossibility since Ll(u - h) = Llu > 0 at a. Then for the proof of the theorem itself, first suppose that Llu ~ 0 throughout D. For any e > 0, note that the function u. defined by u.(z) = u(z) + elzl 2 satisfies Llu.(z) =

J

Subharmonic Functions

101

+ 48 > 0 at each point zED. Hence, by the preliminary observation, u. is sub harmonic in D. The functions u. are a monotonically decreasing family of functions converging to u as 8 tends to zero, so u is subharmonic by Theorem 4. Next suppose that u is subharmonic in D. If ~u < 0 at some point a E D, then by the preliminary observation the function - u is also subharmonic near a; but then ~u = 0 near a, which is a contradiction. Thus, ~u ~ 0 throughout D, and the proof is thereby concluded. ~u(z)

K Pluriharmonic and Plurisubharmonic Functions

From the point of view of complex analysis, harmonic functions in e are particularly interesting as being at least locally the real parts of holomorphic functions. Only a subclass of the harmonic functions in en are locally the real parts of holomorphic functions when n > 1, and it is clearly of some importance to investigate more closely this subclass. It is a consequence of Hartogs's theorem that hoi omorphic functions of several variables can be characterized as those functions that are holomorphic in each variable separately, but there is no corresponding characterization of the real parts of holomorphic functions. For example, the function u(z 1, z2) = Xl Y2 is the real part of a holomorphic function of each variable separately but is easily seen not to be the real part of a holomorphic function of two variables. However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in en through a point A Een in the direction of a vector BEen is the one-dimensional complex submanifold of en described parametrically as {A + tB: t EC}. 1. DEFINITION. A real-valued function in an open subset D ~ en is a pluriharmonic function if it is continuous in D and its restriction to any complex line through any point of D is a harmonic function on that line in D. If f is a holomorphic function in an open subset D ~ en, then the restriction of f to any complex line through any point of D is a holomorphic function on that line; hence, the real part of f is a pluriharmonic function in D. On the other hand, the function U(Zl' Z2) = X 1 Y2 mentioned above is harmonic in e 2 but is not pluriharmonic, since u(t, it) = (Re t)2.

The pluriharmonic functions in an open subset D ~ en are harmonic and hence are of class coo in D. A real-valued function u of class C 2 in D is a pluriharmonic function precisely when aau = 0 throughout D.

2. THEOREM.

Proof. If u is continuous in D and dJ-l is the standard surface measure on the boundary aB(A; r) of any closed ball B(A; r) ~ D, then the spherical average

K PI uri harmonic and Plurisubharmonic Functions

103

loB(A; r)I- 1 faB(A;r) u(Z) dfJ.(Z) can be written as an integral average of the average values of u over the circles oB(A; r) n L for all complex lines L through A; it is not really necessary to know the explicit formula. If u is plurisubharmonic in D, then the restriction ulL is harmonic, so the average value of u over the circle oB(A; r) n L reduces to the value u(A), and hence the spherical average also reduces to the value u(A); as is well known, that ensures that u is harmonic in D. If u is any function of class C 2 in D, then for any complex line {A + tB: t E C} through a point A E D, the complex form of the chain rule for differentiation shows that the Laplacian of the restriction of u to this complex line can be written in the form (1)

If u is a pi uri harmonic function in D, then ~tu(A + tB) = 0 for all A, B, t, with A + tB E D; hence, 02U(Z)/OZj OZk = 0 throughout D. Conversely, if 02U(Z)/OZj OZk = 0 throughout D, then ~tu(A + tB) = 0 for all A, B, t with A + tB E D; hence, u is pluriharmonic in D. Since oau(Z) = Ljk(02U(Z)/OZj Ozk) dZj A k, that suffices to conclude the proof.

az

This shows that the second-order linear differential operator oa plays for pluriharmonic functions the role the operator aplays for holomorphic functions. The pluriharmonic functions are characterized by the system of partial differential equations oau = 0, a more complicated situation than the characterization of harmonic functions by the single partial differential equation ~u = O. When expressed in terms of the underlying real coordinates Xj' Yj in C" = R211 , the system of partial differential equations oau = 0 takes the form (2)

3. THEOREM. The pluriharmonic functions in a simply connected open subset D precisely the real parts of the holomorphic functions in D.

S;

en are

Proof. Suppose u is a pluriharmonic function in D, so that by the preceding theorem u is of class COO and satisfies oau = 0 throughout D. The differential form OU is then a closed hoi om orphic I-form in D, so since D is simply connected, the indefinite integral f(Z) = ou is a well-defined holomorphic function in D. Note that df = ou, so that d(f + 1 - u) = df + dj - du = OU + au - du = 0; but then f + 1 - u = c is a real constant, and therefore u = f + 1 - c = Re(2f - c) is the real part of a holomorphic function in D. The converse was already observed, and that concludes the proof.

n

Since only a subclass of harmonic functions in C" are of particular interest in studying holomorphic functions of several variables, it might be expected that correspondingly only a subclass of subharmonic funCtions in e" are of particular interest in this context. That is indeed the case, as has been made quite evident by

104

Volume I

Function Theory

the pioneering work of Oka and Lelong. The relevant concept is the following one. A mapping u: D -+ [ -00, +(0) defined in an open subset D £; en is a plurisubharmonic function if: (i) u is upper semicontinuous in D and (ii) the restriction of u to any any complex line through any point of D is a subharmonic function on that line in D.

4. DEFINITION.

The parallels between Definitions 1 and 4 are quite apparent. As in the case of subharmonic functions, upper semicontinuity is the natural regularity condition to require in this context, but upper semicontinuity on each complex line is evidently not enough to imply upper semicontinuity in en, so the latter condition must be required separately. The reasons for requiring this form of regularity will become apparent in the subsequent discussion. It should also be pointed out here that the function that is identically equal to - 00 is plurisubharmonic according to this definition, but is sometimes not considered to be plurisubharmonic in other definitions. Most of the general elementary properties of subharmonic functions in e discussed in the last section extend quite trivially to hold for plurisubharmonic functions in en. For convenience the following brief catalog of such properties is included here. Let D be an open subset of en and u, u l , u 2 , ... be mappings from D into +(0). (a) A mapping u is plurisubharmonic in D precisely when it is plurisubharmonic in an open neighborhood of each point of D. (b) If u is plurisubharmonic in D and ifu(A) = SUPZEDU(Z) for some point A e D, then u is constant in the connected component of D containing A. (c) If u is plurisubharmonic in D, then u(A) = lim SUPZ-+AU(Z) for each point AeD. (d) If u is plurisubharmonic in D and if rP: IR -+ IR is a convex and monotonically increasing function, then the composition rP 0 u is also plurisubharmonic in D, where rP( -(0) = lim x -+_ oo rP(x). (e) If u l , U 2 are plurisubharmonic in D and if c l , C2 are positive real numbers, then the mapping u defined by u(Z) = ClUl(Z) + C 2 U 2 (Z) is also plurisubharmonic inDo (f) If u l , U2 are plurisubharmonic in D, then the mapping u defined by u(Z) = sup(u l (Z), u 2 (Z)) is also plurisubharmonic in D. (g) If {u v} is a monotonically decreasing sequence of plurisubharmonic functions in D, then the mapping u defined by u(Z) = lim v uv(Z) is also plurisubharmonic in D. (h) Any pluriharmonic function is also plurisubharmonic, and if both u and - u are plurisubharmonic, they are actually pluriharmonic.

5. THEOREM.

[ -00,

Proof. All but parts (c) and (g) of this theorem follow almost immediately from the definitions and Theorem 12. Part (c) follows from part (b) as in the proof of the

K Pluriharmonic and Plurisubharmonic Functions

105

corresponding assertion in Theorem J2, while part (g) follows directly from Theorem J4. The first part of the preceding theorem can be rephrased as the assertion that plurisubharmonicity is a local property; as in the case of subharmonicity, that is a very useful observation and will be used repeatedly. Part (b) is a maximum theorem for plurisubharmonic functions; as noted in the discussion ofthe corresponding result

for subharmonic functions, this is analogous to but slightly weaker than the usual maximum modulus theorem for holomorphic functions. Parts (e) and (h) can be reinterpreted as the assertions that the set of plurisubharmonic functions in D form a convex cone, with the pluriharmonic functions in D forming the maximal linear subspace contained in that cone.

The integrability properties of subharmonic functions in e discussed in the preceding section also extend to hold for plurisubharmonic functions in en, but with some restrictions and a bit more effort. It follows immediately from Theorems J5 and J6 that if u is plurisubharmonic in an open subset D ~ en and if the restriction of u to some complex line L = {A + tB: t E C} is not identically equal to - 00 in a connected component U of D n L, then u is locally Lebesgue integrable in U and is integrable over every circle contained in U. The restriction of u to some line L may of course be identically equal to -00 even when u is not identically equal to -00 in D, so some care must be taken in applying the preceding observations. However, the restriction of u is not identically equal to - 00 on almost all lines in en, as a consequence of the following natural extension of Theorem J5.

6. THEOREM.

If u is plurisubharmonic and not identically equal to - 00 in a connected open subset D ~ en, then u is locally Lebesgue integrable in D. Hence, {Z ED: u(Z) = - 00 } is a subset of D of Lebesgue measure zero.

Proof. The proof is essentially a repetition of the proof of Theorem J5, the corresponding result for sub harmonic functions. If u is plurisubharmonic in D and if A(A; R) ~ D, then the restriction U(Zl' a2"'" an) is subharmonic in A(a 1 ; rd, so as in the proof of Theorem J5, u(A)

~ I~(al; rdl-

1

r

J.~(al;rl)

U(Zl' a2' ... , an) dV(zd

For a fixed Zl there is a similar inequality for U(Zl' Z2"'" zn) viewed as a subharmonic function of Z2 in A(a2; r2), and so on. Altogether, u(A)

~ I~(A; R)I- 1

f

~~

u(Z) dV(zd'" dV(zn) =

I~(A; R)I- 1

r

J~~

u(Z) dV(Z)

where dV(Z) is Lebesgue measure in en, the equality of the iterated and multiple integral following immediately from Fubini's theorem for positive functions

106

Volume I

Function Theory

(Tonelli's theorem), since u is bounded from above on the compact set A(A; R). This formula leads immediately to the useful preliminary observation that if u(A) #- - 00 and A(A; R) £: D, then u is Lebesgue integrable over A(A; R). Now let E be the subset of D consisting of those points A E D such that u is integrable over an open neighborhood of A. Clearly E is an open subset of D, and since u is not identically equal to - 00 by hypothesis, it follows from the preliminary observation that E is nonempty. On the other hand, if BED - E and if A(B; 2R) £: D, it also follows from the preliminary observation that u is identically equal to - 00 in A(B; R); for if u(A) > - 00 at some point A E A(B; R), then A(A; R) £: A(B; 2R) £: D and u is integrable over A(A; R), but since BE A(A; R), it follows that BEE, a contradiction. Thus, A(B; R) £: D - E, so that D - E is also open. Since D is connected, necessarily E = D, and that suffices to conclude the proof. Next for the special case of smooth plurisubharmonic functions, there is a natural extension of the characterization of smooth subharmonic functions given in Theorem J8, for which extension it is convenient to introduce some further notation.

7. DEFINITION.

If u is a function of class C2 in an open subset D £: a point ZED is the Hermitian form

for any vectors B, C E

en, the Levi form of u at

en. For simplicity of notation also set Lu(Z; B, B) = Lu(Z; B).

The Levi form is thus the Hermitian form naturally associated to the complex Hessian matrix a 2 u(Z)/az; Ozj of the function u, so it is the Hermitian version of the skew-Hermitian form also associated to that Hessian matrix. The usefulness of this form in studying holomorphic functions of several variables was made clear in the work of E. E. Levi.

aau

8. THEOREM.

A real-valued function of class C 2 in an open subset D £: en is plurisubharmonic in D precisely when its Levi form is positive semidefinite at each point of D.

Proof. If u is a function of class C 2 in D, then for any complex line {A + tB: t E C} through a point A EDit follows from the complex form of the chain rule for differentiation as in (1) that Atu(A + tB)lt=o = Lu(A; B). It is then an immediate consequence of Theorem J8 that u is plurisubharmonic in D precisely when Lu(A; B) ~ 0 for all points A E D and all vectors BEen-hence, precisely when the Levi form of u is positive semidefinite at each point of A. That suffices for the proof.

A convenient method for deriving some additional properties of plurisubharmonic functions is to show that sufficiently smooth plurisubharmonic functions

K Pluriharmonic and Plurisubharmonic Functions

107

have the desired properties and then to approximate general plurisubharmonic functions by smooth plurisubharmonic functions in such a way that the properties of interest also hold in the limit. One ofthe standard smoothing operators in analysis provides a useful approximation for such purposes. To introduce this operator choose once and for all a real-valued COO function a in the complex plane such that a(z) depends only on Izl, a(z) ~ 0 for all z E e, a(z) = 0 whenever Izl ~ 1, and Ie a(z) dV(z) = 1 where dV(z) is Lebesgue measure on e. Then set a(Z) = a(z 1)' .. a(zn) in en, and let dV(Z) be Lebesgue measure on en. For any locally Lebesgue integrable mapping u: D ~ [ -00, +00) defined in an open set D ~ en and any real constant e > 0, let

9. DEFINITION.

u£(Z)

= [

u(Z

+ eW)a(W) dV(W)

(3)

JWEC n

whenever Z is contained in the open subset D£

= {Z E D: t5D (Z) > e}

~ D

The notation t5D is that introduced in Definition G3.1f ZED., then Li(Z; e)~D, and since the integration in (3) can be restricted to the closed polydisc Li(O; 1) containing the support of a and Z + eW E Li(Z; e) ~ D whenever WE Li(O; 1), it is clear that the function u£ is well defined in D. By the obvious change of variables, the integral (3) can be rewritten u£(Z) = e- 2n

[

JWECn

u(W)a (W -

e

Z)

dV(W)

(4)

It is clear from the latter formula that u£ is actually a function of class Coo in D. That the functions u£ do generally approximate u is a well-known result, but for the sake of completeness a proof will be appended here.

10. LEMMA. If u: D ~ IR is continuous, the Coo functions u£ converge to u uniformly on any compact subset of D as e tends to zero. If u: D ~ [ - 00, + 00] is locally Lebesgue integrable, the Coo functions u£ converge to u in L1-norm on any compact subset of D as e tends to zero. If u: D ~ [ - 00, + 00] is locally square-integrable, then it is also locally integrable and the Coo functions u£ converge to u in L 2- norm on any compact subset of D as e tends to zero. Proof. If u is continuous on D, it is uniformly continuous on any compact subset of D. Given a compact ~ubset K ~ D and a constant '1 > 0, choose t5 > 0 sufficiently small that K' = UZEKA(Z; 0) ~ D and lu(Zd - u(Z2)1 < '1 whenever Zl and Z2 are any two points of the compact set K' such that IIZl - Z211 < 0. Then whenever o ~ e < 0 and Z E K, it follows that

108

Volume I Function Theory

lu.(Z) - u(Z)1 =

r

(u(Z

r

'1U(W) dV(W)

I

+ eW) -

u(Z»t5(W) dV(W)1

JWe4(O;1)

~

= '1

JWe4(O;1)

as desired. Next suppose that u is locally Lebesgue integrable in D, that V is an open subset of D such that [j is a compact subset of D, and that '1 > 0 is any given positive constant. Choose an open neighborhood of V of [j such that V is also a compact subset of D, and choose a continuous function v of compact support in V such that IIu - vII 1. v < '1/3, where 11'11 1. V denotes the usual L 1- norm on V. That there exists such a function v is a well-known result from the theory of the Lebesgue integral. Whenever e > 0 is sufficiently small that [j ~ then the functions u. and v. are well defined in V and

v.,

I u. -

f ~f

v. I 1. U =

Iu.(Z) -

v.(Z)1 dV(Z)

ZeU

r

lu(Z

+ eW) -

v(Z

+ eW)lu(W) dV(W) dV(Z)

Ze U JWe4(O; 1)

The order of integration can be interchanged by Fubini's theorem, and since [j ~ V. and the last integral is increased by integrating over a larger set, it follows that

IIu.-v.II1.U~

r

IIu-v ll 1 • v u(W)dV(W) 0 is sufficiently small that [j ~ then u. and v. are well defined in V and

v.,

K Pluriharmonic and Plurisubharmonic Functions

Ilu. -

f ~f

v.II~,u =

109

lu.(Z) - v.(ZW dV(Z)

ZeU

r

lu(Z

+ eW) -

v(Z

+ eWWa(W) dV(W) dV(Z)

ZeU JWe4(0;1)

since by Schwarz's inequality lu.(Z) - v.(Z) I

~

f

lu(Z

+ eW) -

v(Z

+ eW)la(W)1/2·a(W)1/2·dV(W)

4(0;1)

~ {f

lu(Z

+ eW) -

v(Z

+ eWWa(W) dV(W)}1/2

4(0;1)

x

{f

a(W) dV(W)}1/2

4(0;1)

The order of integration can be interchanged by Fubini's theorem, and since [J it follows that

lIu. -

v.lltu

~

r

JWe4(0;1)

lIu _

vll~,va(W)dV(W)
0 such that u~ + tv is plurisubharmonic on UA whenever It I < ~A' and then of course u + tv = (u~ + tv) + u~ is also plurisubharmonic in Uk Finitely many such neighborhoods UA J_ will cover the compact support of v, so for ~v = infJ'~A_J > 0, it follows that u + tv is plurisubharmonic on the support of v and hence in all of D whenever It I > ~v. That suffices to conclude the proof. A mapping u: D - [ - 00, + 00) defined in an open subset D ~ plurisubharmonic in D if and only if:

3. THEOREM.

(i) u is locally Lebesgue integrable in D, and for each point A u(A)

1

= lim IA(A; 8)1- 1 .-+0

a(A;.)

E

en is strictly

D,

u(Z) dV(Z)

(ii) for any open subset U ~ D such that [J is a compact subset of D, there is a constant (ju > 0 such that

L

u(Z)· Lv(Z; A) dV(Z)

~ (ju1l A 1I2 So v(Z) dV(Z)

whenever v is a nonnegative Coo junction with support contained in U and A

E

en.

120

Volume I

Function Theory

Proof. First suppose that u: D --+ [ - 00, + 00) is a mapping satisfying the two conditions of this theorem. In order to show that u is strictly plurisubharmonic in D, it is enough by Theorem 2 to show that for any COO function v with compact support KeD there is a constant (jv > 0 such that u + tv is plurisubharmonic in D whenever - (jv < t < (jv, since u is not identically equal to - 00 in any connected component of D by condition (i). Choose an open neighborhood U of K such that [J is also a compact subset of D, let (ju > 0 be the constant associated to the set U so that condition (ii) holds, and let Av > 0 be any constant such that ILv(Z; A)I ~ Av II A 112 for all ZED and A E en. Observe that there obviously exists such a constant, since v is Coo with compact support in D. The constant (jv = A;l(jU then has the desired property; that is, u + tv is plurisubharmonic in D whenever - (jv < t < (jv' To verify this note first that conditions (i) and (ii) imply that u itself is plurisubharmonic in D as a consequence of Theorem K15. The function u + tv coincides with u in D - K and hence is plurisubharmonic there, so to show that it is plurisubharmonic in D, it is enough just to show that it is plurisubharmonic in U. For this purpose the plurisubharmonicity criterion of Theorem K15 will again be used. It is clear thatu + tv satisfies condition (i) of Theorem K15, since u does so by hypothesis and v is Coo in U. As for condition (ii) of Theorem K 15, if w is any nonnegative Coo function having compact support in U and A is any vector in C", then recalling Lemma K14 shows that

In

[u(Z)+tV(Z)]' Lw(Z; A) dV(Z) =

In

u(Z)· Lw(Z; A) dV(Z)+t

In

w(Z)· Lv(Z; A) dV(Z)

~ (ju11 A 112 In w(Z) dV(Z) -ltlAvll A l12 In w(Z) dV(Z) by condition (ii) of this theorem and the definition of the constant Av. The last expression above is ~O whenever It I < A;l(ju = (jv; hence, u + tv is plurisubharmonic in U whenever It I < (jv as desired. Next suppose that u is a strictly plurisubharmonic function in D. The function u is in particular plurisubharmonic in D and is not identically equal to - 00 in any connected complement of D, so it follows from Theorem K15 that u satisfies condition (i) of the present theorem. As for condition (ii), consider an open subset U ~ D such that [J is a compact subset of D. In an open neighborhood UA of any point A E [J, the restriction ul UA = u~ + u~, where u~ is a function of class C 2 with Levi form strictly positive definite at each point of UA and u~ is plurisubharmonic in Uk After shrinking the neighborhood UA if necessary, it can be assumed that the minimum eigenvalue of the Levi form of u~ is bounded away from zero on UA and hence that there is a constant (jA > 0 for which Lu~(A; B) ~ (jA IIBII2 whenever Z E UA and 1!. E C". Choose finitely many such neighborhoods UAJ covering the compact set U, and let (ju = in~ (jA > O. If v is a nonnegative Coo function with support contained in U, write v = Vj' where Vj is a nonnegative Coo function with , by using a Coo partition of unity for the covering {UA .} of support contained in U A _ J J U as usual. Then from Lemma K14 it follows that

L

l

=~ J

i

uAj

Special Classes of Plurisubharmonic Functions

[Vj(Z)·

121

Lu~/Z; B) + u~JZ)· LVj(Z; B)] dV(Z)

~ ~ b )B11 2LA. viZ) dV(Z) A

}

~ bu IIBI12

L

v(Z) dV(Z)

for SUA u~/Z)· LVj(Z; B) dV(Z) ~ 0 by Theorem K15, since u~/Z) is plurisubharmonic.} Thus, u satisfies condition (ii) also, and that suffices to conclude the proof of the theorem. The preceding theorem provides a characterization of strictly plurisubharmonic functions paralleling the characterization of ordinary plurisubharmonic functions given in Theorem K15. Condition (i) is one of the regularity conditions satisfied by all plurisubharmonic functions nowhere identically equal to - 00, while condition (ii) is just the condition that the Levi form interpreted in the sense of distributions is strictly positive definite everywhere. In some ways this is the most natural characterization of general strictly plurisubharmonic functions, while the characterization given in Theorem 2 is perhaps the most functionally useful. The special role played by strictly plurisubharmonic functions will become more evident in the course of the subsequent discussion, so nothing will be said about that here. However, it is useful to have available a catalog of some of the operations on plurisubharmonic functions that preserve the subclass of strictly plurisubharmonic functions. It is evident that no limiting processes can be expected to preserve this subclass, and that in particular the strictly plurisubharmonic functions do not form a closed subset of the space of plurisubharmonic functions in the topology of local U-convergence; but most purely algebraic operations do generally preserve this subclass.

4. THEOREM.

If u, v are strictly plurisubharmonic functions in an open subset D s; a, b are nonnegative real constants, then (a) au

en and

+ bv is strictly plurisubharmonic in D; and

(b) sup(u, v) is strictly plurisubharmonic in D. Proof. (a) It is obvious that the first assertion is true for strictly plurisubharmonic functions of class C 2 • In general, in an open neighborhood UA of any point A ED, the functions u, v can be written ul UA = u~ + u~, vi UA = v~ + v~, where u~, v~ are strictly plurisubharmonic functions of class C 2 and u,;" v';' are plurisubharmonic functions; but then (au + bv)IUA = (au~ + bv~) + (au';' + bv~), where au~ + bv~ is

122

Volume I

Function Theory

strictly plurisubharmonic of class C 2 while au~ + bv~ is plurisubharmonic; hence, au + bv is strictly plurisubharmonic as desired. (b) It follows from Theorem 2 that for any COO function w with compact support in D, there is a constant t5w > 0 such that u + tw, v + tw are plurisubharmonic in D whenever - t5w < t < t5w ; but then by Theorem K5(f) the function sup(u, v) + tw = sup(u + tw, v + tw) is also plurisubharmonic in D, so from Theorem 2 again it follows that sup(u, v) is strictly plurisubharmonic as desired. That suffices to conclude the proof. If D £; em and E £; en are connected open subsets, F: D --+ E is a nonsingular holomorphic mapping from D into E, and u is a strictly plurisubharmonic function in E, then the composite mapping u 0 F is either identically equal to - 00 or a strictly plurisubharmonic function in D.

5. THEOREM.

Proof. Suppose that u 0 F is not identically equal to -00 in D. For any point A E D, it follows from Definition 1 that in an open neighborhood U of the point F(A) E E, the function u can be written as the sum ul U = u' -\- u", where u' is a strictly plurisubharmonic function of class C 2 in U and u" is plurisubharmonic in U. It follows directly from the complex form of the chain rule for differentiation that the Levi form of the composite function u' 0 F is given by L(u' 0 F)(Z; B) = Lu'(F(Z); JF(Z)· B) for any point Z E F-1(U), and since Lu' is positive definite and the Jacobian matrix JF(Z) is nonsingular by hypothesis, it follows that L(u' 0 F) is also positive definite and hence that u' 0 F is strictly plurisubharmonic in F- 1 (U). On the other hand, u" 0 F is plurisubharmonic in F-1(U) by Theorem K12; hence, u 0 F = u' 0 F + u" 0 F is also strictly plurisubharmonic in F-1(U), and the proof

is thereby concluded. It is clear that the non singularity of the mapping F is a necessary hypothesis in the preceding theorem, since it is even necessary in the special case of C2 strictly plurisubharmonic functions. However, it still follows just as for ordinary plurisubharmonic functions that the class of strictly plurisubharmonic functions is preserved by biholomorphic mappings. It is thus possible to speak of strictly plurisubharmonic functions on an arbitrary complex manifold. Furthermore, if u is a strictly plurisubharmonic function in an open subset D £; en and M is a connected complex

submanifold of D, then the restriction ulM is either identically equal to strictly plurisubharmonic function on M.

-00

or a

The other special class of functions to be considered here is a properly larger class of functions than the plurisubharmonic functions, consisting of functions that are equal almost everywhere to a plurisubharmonic function but that may not be plurisubharmonic because they fail to be upper semicontinuous. Such functions do arise in practice, and an examination of them here may serve to clarify the role that upper semicontinuity plays in the theory of plurisubharmonic functions. It is convenient to begin the discussion by recalling some more general properties of upper semicontinuous functions. For any mapping u: D --+ [ - 00, -\- 00] defined in an open set D s;;; en, the upper envelope of u is the mapping u*: D --+ [ - 00, -\- 00] defined by

L Special Classes of Plurisubharmonic Functions

123

u*(A) = sup (U(A), lim sup U(Z)) z~A

= lim ( sup U(Z)) e~O

(1)

Z E B(A;e)

The upper envelope has the following properties, familiar from elementary analysis.

6. LEMMA.

If u*, v* are the upper envelopes of mappings u, v: D -. [ - 00, + 00]: (a) u(A) ~ u*(A) at each point A E D. (b) If u(A) ~ v(A) at each point A ED, then also u*(A) ~ v*(A). (c) If u is locally bounded from above, then u* is upper semicontinuous. (d) If u is upper semicontinuous, then u = u*.

Proof. (a), (b) These assertions are trivial consequences of the definition (1) of the upper envelope. (c) If u is locally bounded from above, it follows immediately from (1) that u*(A) < 00 at each point A E D and hence that u* is actually a mapping u*: D -. [ - 00, + (0). If u*(A) < r at some point A E D, it further follows from (1) that there is a value t: > 0 for which SUPZEB(A;e)U(Z) < r, and then it is clear from (1) again that u*(Z) < r at each point Z E B(A; t:). Thus, {Z ED: u*(Z) < r} is an open subset of D for any real number r, so that u* is upper semicontinuous as desired. (d) If u is upper semicontinuous, then u(A) ~ lim SUPZ~A u(Z), and it then follows from (1) that u*(A) = u(A) at each point A E D. That suffices to conclude the proof.

Note that if u: D -. [ - 00, + (0) is a mapping that is locally bounded from above, if v: D -. [-00, +(0) is an upper semicontinuous mapping, and if u(Z) ~ v(Z) at each point ZED, then it follows immediately from parts (a), (b), and (d) of the preceding lemma that u(Z) ~ u*(Z) ~ v*(Z) = v(Z) at each point ZED, while u* is itself upper semicontinuous as a consequence of part (c) of that lemma. Therefore, the upper envelope of a mapping u: D -. [ - 00, + (0) that is locally bounded from above can be characterized as the least upper semicontinuous mapping u*: D -. [ - 00, + (0) such that u ~ u*. In terms of this construction then, introduce the following special class of functions.

A mapping u: D -. [-00, +(0) in an open subset D S; en is called a nearly if its upper envelope u* is a plurisubharmonic function in D and u(Z) = u*(Z) for all points ZED outside a subset of D of Lebesgue measure zero.

7. DEFINITION.

plurisubharmonic function

It is clear that the condition that a mapping be a nearly plurisubharmonic function is a local condition. Any plurisubharmonic function is obviously also a nearly plurisubharmonic function, so that the class of nearly plurisubharmonic functions is an extension of the class of plurisubharmonic functions. Note that a nearly plurisubharmonic function can be viewed as a mapping derived from a

124

Volume I

Function Theory

plurisubharmonic function by decreasing the values of the latter function on a set of measure zero. Indeed, any such operation necessarily leads to a nearly plurisubharmonic function as a consequence of the following observation. If v is a plurisubharmonic function in an open set D ~ en, and if u: D --+ [-00, +(0) is a mapping such that u(Z) ~ v(Z) for all points ZED while u(Z) = v(Z) for almost all points ZED in the sense of Lebesgue measure, then u is a nearly plurisubharmonic function in D.

8. THEOREM.

Proof. It suffices to show that the hypotheses of this theorem imply that u* = v; for v is plurisubharmonic, and it follows from Lemma 6 that u(Z) ~ u*(Z) ~ v*(Z) = v(Z) for all points ZED and hence that u(Z) = u*(Z) almost everywhere in D. On the one hand, u*(Z) ~ v(Z) for all points ZED as just observed. On the other hand, if u*(A) < v(A) at some point A ED, then it follows immediately from the definition (I) of the upper envelope that there are constants 8 > 0, () > 0 such that supz e B(A;d) u(Z) = v(A) - 8. But since v is plurisubharmonic and v = u almost everywhere in D, it then follows from Theorem KI5 that

r-+O

J.~(A;r)

r

v(Z) dV(Z)

lim IA(A; r)I- 1 r-+O

Jr

u(Z) dV(Z)

v(A) = lim IA(A; r)I- 1

=

~

v(A) -

L1(A;r)

8

a contradiction. Therefore, u* = v as desired, and the proof is thereby concluded. Let u 1 , u 2, ... be nearly plurisubharmonic functions in an open subset D ~ en. (a) If ai' a2 are nonnegative real numbers, then al Ul + a2 U2 is also nearly plurisubharmonic in D. (b) If the functions U v form a monotonically decreasing sequence, then the limit function u = limv U v is also nearly plurisubharmonic in D. (c) If the functions U v are uniformly bounded from above on any compact subset of D, then the functions u, v defined by u(Z) = supv uv(Z) and v(Z) = lim sUPv-+oo uv(Z) are also nearly plurisubharmonic in D.

9. THEOREM.

Proof. (a) If u = a 1 u 1 + a2u2, it is clearfrom the definition (1) of the upper envelope that u(Z) ~ u*(Z) ~ a 1 uT(z) + a 2u!(Z) for all points ZED. But a l uT + a2u! is plurisubharmonic in D while u(Z) = a 1 uT(Z) + a 2u!(Z) for almost all points ZED since the functions Ut> U 2 are nearly plurisubharmonic, so it follows immediately from Theorem 8 that u is nearly plurisubharmonic in D. (b) It is obvious from Lemma 6 that the functions u~ also form a monotonically decreasing sequence, and since each ofthese functions is plurisubharmonic, the limit w = limv u~ is plurisubharmonic by Theorem K5(g). But clearly u(Z) ~

L Special Classes of Plurisubharmonic Functions

125

w(Z) for all points ZED, while u(Z) = w(Z) for almost all points ZED, so it follows immediately from Theorem 8 that u is nearly plurisubharmonic in D. (c) It can obviously be assumed without loss of generality that none of the functions u. is identically equal to - OCJ on any connected component of the set D. The same is also true of the pI uri sub harmonic functions u~, which are then locally integrable in D by Theorem K6. For the auxiliary function w defined by w(Z) = suP. u~(Z), it is clear that u(Z) ~ w(Z) for all points ZED, while u(Z) = w(Z) for almost all points ZED. The functions u~ are evidently locally uniformly bounded from above, so their supremum w is locally bounded from above, and since w ~ ui, the function w is necessarily locally integrable as well. The sequence of plurisubharmonic functions w. defined by w.(Z) = sup(ui(Z), ... , u~(Z» converges pointwise to w, and since ui ~ w. ~ w, it follows readily from the Lebesgue dominated convergence theorem that the sequence of functions w. also converges to w in the topology of local L l-convergence. Therefore, from Corollary K 17 the function w is equal almost everywhere in D to a plurisubharmonic function w. Since u~(Z) ~ w(Z) = w(Z) for almost all points ZED while both u~ and ware plurisubharmonic, it follows from condition (i) of Theorem K15 that actually u~(Z) ~ w(Z) for all points ZED. Consequently, w(Z) = suP. u.(Z) ~ w(Z) for all points ZED, while w(Z) = w(Z) for almost all points ZED. Combining this with the relation between the functions u and w observed above shows that u(Z) ~ W(Z) for all points ZED while u(Z) = W(Z) for almost all points ZED. It then follows immediately from Theorem 8 that u is nearly plurisubharmonic in D. Finally, since v(Z) = lim sup .....oo u.(Z) = lim .....oo v.(Z) where v.(Z) = sup(u.(Z), U.+ l (Z), ... ), the functions v.(Z) are nearly plurisubharmonic by what has just been proved, and since they form a monotonically decreasing sequence, it follows from part (b) of this theorem that their limit v is also nearly plurisubharmonic in D as desired. That suffices to conclude the proof.

The first assertion of the preceding theorem shows that the nearly plurisubharmonic functions form a convex cone, while the second assertion shows that the set of nearly plurisubharmonic functions is closed under one of the basic limiting processes known to preserve plurisubharmonic functions. The third assertion is really the most interesting part of the theorem and is essentially the reason for introducing this class of functions. It shows that the nearly plurisubharmonic functions are closed under more general operations than those that preserve the ordinary plurisubharmonic functions. Whereas the supremum of any finite collection of plurisubharmonic functions is necessarily plurisubharmonic, that is not necessarily the case for the supremum of a countably infinite collection of plurisubharmonic functions, even when they are locally uniformly bounded from above; the latter supremum may be only nearly plurisubharmonic. Actually for the case of nearly plurisubharmonic functions, it is not even necessary to restrict to a countably infinite collection of functions for the preceding theorem to hold. A special case of a general observation of Choquet leads immediately to the following result. 10. THEOREM. If rUt} is any collection of nearly plurisubharmonic functions in an open subset D £; en, and if these functions are uniformly bounded from above on any

126

Volume I Function Theory

compact subset of D, then the function u defined by u(Z) = SUPt ut(Z) is nearly plurisubharmonic in D. Proof. Let {Up} be a countable basis for the open subsets of D such that each set Up is a compact subset of D, and for each integer v ~ 1, let up. be one of the functions {u t } for which

sup up.(Z) ZeU.

~

1 1 sup ut(Z) - - = sup u(Z) - V

t,ZeU.

(2)

V

ZeU.

The function v defined by v(Z) = sUPP , ' up.(Z) is nearly plurisubharmonic in D by Theorem 9(c), and obviously v(Z) ~ u(Z) and hence v*(Z) ~ u*(Z) for all points ZED. In order to complete the proof, it is only necessary to show that v*(Z) = u*(Z) for all points ZED; for v* is plurisubharmonic in D, and since v(Z) ~ u(Z) ~ u*(Z) = v*(Z) at all points ZED, while v(Z) = v*(Z) at almost all points in D, then u(Z) = u*(Z) at almost all points in D. If, in contradiction to the desired result, v*(A) = u*(A) - 2e < u*(A) for some point A ED and some e > 0, then it follows directly from the definition (1) ofthe upper envelope v* that sUPZeU v(Z) < u*(A)-e for one of the basic open sets Up containing A and hence that ·suPzeu. up.(Z) < u*(A) - e for every integer v. But then using (2) and recalling the definition (1) of the upper envelope u* show that u*(A) - e > sup up.(Z) ZeU.

~

1 1 sup u(Z) - - ~ u*(A) - ZeU.

V

V

for every integer v, and that is impossible whenever v is so large that l/v < e. Therefore, v*(Z) = u*(Z) for all points ZED, and the proof is thereby concluded. It is perhaps worth pointing out explicitly that if, for example, {u t } is a collection of nearly plurisubharmonic functions in an open set D £; en indexed by the points t in an open set T £; em, then for any point to E T, the function Vto defined by vto(z) = lim SUPt-+to ut(Z) is also nearly plurisubharmonic in D; for Vto = lim.-+o vto" where vto.(Z) = SUPteB(to;') ut(Z), the function Vto' is nearly plurisubharmonic in D by Theorem 10, and since the functions Vto' are monotonically decreasing as e tends to zero, the limit function Vto is nearly plurisubharmonic in D by Theorem 9(b).

M Pseudoconvex Subsets of

en

Plurisubharmonic functions have proved quite useful in the further investigation of domains of holomorphy and related topics, following the pioneering work of K. Oka. Recall from section G that an open subset D s;;; en is a domain of holomorphy precisely when it is holomorphically convex, and that the latter condition is expressed in terms offunctions ofthe form If I where f is holomorphic in D. Now it is a familiar and obvious consequence of the Cauchy integral formula that the absolute value u = If I of a holomorphic function f of a single variable satisfies the integral inequality J(2) and hence is a subharmonic function. And since the restriction of a holomorphic function of several variables to any complex line is a holomorphic function of one variable, it is evident that the absolute value If I of a holomorphic function f of several variables is a plurisubharmonic function. That suggests looking at corresponding notions of convexity of domains defined in terms of plurisubharmonic functions, as a fairly natural generalization of holomorphic convexity. This suggestion turns out to be very fruitful indeed, as will become apparent from the subsequent discussion. Perhaps the easiest place to begin such an investigation is with the following. An open subset D S;;; en is pseudoconvex if for any compact subset K S;;; D the set KD = {A ED: u(A) ~ SUPZeKU(Z) whenever u is a continuous plurisubharmonic function in D} is also compact. The set KD is the plurisubharmonically convex hull of KinD.

1. DEFINITION.

This definition is parallel to Definition G2, the definition of a holomorphically convex set and the holomorphically convex hull of a compact subset, with continuous plurisubharmonic functions in place of the moduli of holomorphic functions. Note that for any open subset D s;;; en and any compact subset K s;;; D, the plurisubharmonically convex hull KD is a bounded subset of en, since the plurisubharmonic functions IZjl, where Zj are the coordinate functions in en, are bounded on KD by their maximal values on K. Note further that the subset KD is always a relatively closed subset of D, since the functions u considered in the definition of the subset KD are assumed to be continuous. That is a convenience, simplifying the discussion somewhat, and it is for that reason that continuity was required in Definition 1; continuity is not really essential, though, as will later be seen. In view

128

Volume I

Function Theory

of the preceding obser~ations, the condition that KD be compact reallylust amounts to the condition that KD be a closed subset ofcn, or equivalently, that KD be disjoint from an open neighborhood of the boundary oD of D. Since the absolute values of holomorphic functions are always plurisubharmonic, it is apparent that the plurisubharmonically convex hull KD and the holomorphically convex hull KD of a compact subset K of any open set D ~ C" are related by KD ~ K D. If Dis holomorphically convex, then KD is a compact subset of D, and since K Dis a relatively closed subset of K D, necessarily K Dis also compact; thus, a holomorphically convex set is pseudoconvex. The importance of the notion of pseudoconvexity in studying holomorphic functions arises firstly from the fact that pseudoconvexity is actually equivalent to holomorphic convexity, and secondly from the observation that pseudoconvexity is really a much easier property to handle than holomorphic convexity because of the greater abundance and flexibility of plurisubharmonic than of holomorphic functions. The theorem that a pseudoconvex subset ofcn is holomorphically convex is quite nontrivial and will be proved later, after the introduction of some more machinery. The present section will be devoted to a discussion of pseudoconvex sets for their own sake, examining the properties of these sets that follow rather directly from the results about plurisubharmonic functions derived in the preceding sections. These properties then carry over immediately to domains ofholomorphy, after it is proved that a pseudoconvex set is holomorphically convex. There are a variety of useful alternative characterizations of pseudocdnvex sets, exhibiting a corresponding variety of other properties of such sets. The two characterizations to be considered next are of a rather geometric form, and among other things indicate the motivation for the terminology. One of them involves for an open set D ~ cn the distance functions d Dand bD,R of Definition G3. A standard condition that D be a convex subset of C" = 1R211 in the ordinary sense is that -log dD be a convex function of D; what will be demonstrated is that D is pseudoconvex precisely when -log dD or -log bD,R is plurisubharmonic. The other characterization refers to the continuity theorem, Theorem D6. That theorem asserts that a function holomorphic in an open neighborhood of an annulus (A(A'; R') - ~(A'; S')) x A" ~ .C"· X en, where A(A'; S') ~ ~(A'; R'), and also holomorphic in open neighborhoods of polydiscs A(A'; R') x A~, where A;' -+ A", must extend to a function holomorphic in an open neighborhood of the limit polydisc A(A'; R') x A". So if D ~ C" is a domain of holomorphy, then it must be impossible to fit such a family of polydiscs into C" in such a way as to lead to holomorphic extensions beyond D. Perhaps the easiest way to use the continuity theorem in this reverse direction is in the following terms. 2. DEFINITION. An open subset D ~ C" is pseudoconvex in the sense of Hartogs if whenever !:: [0, 1] x A(O; 1) -+ C" is a continuous mapping from the compact subset [0, 1] x ~(O; 1) ~ IR x C into en such that F is holomorphic in A(O; 1) for each fixed point of [0, 1] and that F(t, z) E D if either 0 ~ t < 1, Z E A(O; 1) or t = 1, Z E o~(O; 1), then F([O, 1] x A(O; 1)) ~ D.

In other words, in a suggestive but less precise reformulation, D is pseudoconvex in the sense of Hartogs if for any continuous family of holomorphic discs

M

Pseudoconvex Subsets of C·

129

en for 0 ~ t ~ 1, whenever .1, ~ D for 0 ~ t < 1 and O~1 ~ D, then .11 ~ D. Pseudoconvexity in the sense of Hartogs is then equivalent to ordinary pseudoconvexity, as follows. ~, ~

3. THEOREM. (i) (ii) (iii) (iv) (v)

For any open subset D

~

en the following conditions are equivalent:

D is pseudoconvex. D is pseudoconvex in the sense of H artogs.

The function -log dD is plurisubharmonic in D. For any polyradius R, the function -log t5D ,R is plurisubharmonic in D. There is a polyradius R such that the function -log t5D ,R is plurisubharmonic in D.

Proof. The first step is to show that condition (i) implies condition (ii). Suppose therefore that Dis pseudoconvex and that F: [0, 1) x .1(0; 1] --+ en is a continuous mapping such that F is holomorphic in ~(O; 1) for each fixed point of [0, 1] and that F(t, z) E D if either t E [0, 1), z E .1(0; 1) or t = 1, z E o~(O; 1); what must be shown is that F([O, t] x .1(0; 1)) ~ D. Now the image F([O, t] x O~(O; t)) = K is a compact subset of D, so its plurisubharmonically convex hull KD must also be a compact subset of D. For any fixed t E [0, 1) and any continuous plurisubharmonic function u in D, it follows from Theorem K12 that the composition u(F(t, z)) is subharmonic in z E ~(O; 1). Since this composition is also continuous in z E .1(0; 1), it then follows easily from the maximum theorem for subharmonic functions, Theorem J2(b), that u(F(t, a)) ~ sUPzeo.1(O; 1) u(F(t, z)) for any point a E .1(0; 1). Thus, u(F(t, a)) ~ supz eK u(Z), and since this holds for all u, necessarily F(t, a) E KD' But that means that F([O, 1) x .1(0; 1)) ~ KD, and since KD is compact and F is continuous, it then follows that F([O, 1] x .1(0; 1)) ~ KD ~ D as desired. The next step is to show that condition (ii) implies conditions (iii) and (iv); essentially the same proof gives both implications. Suppose therefore that D is pseudoconvex in the sense of Hartogs, and as a convenient abbreviation let d stand for either dD or t5D ,R in this portion ofthe proof. Since d is easily seen to be continuous in D, in order to show that d is plurisubharmonic it is sufficient just to verify that its restriction to any complex line through any point of Dis subharmonic. For that purpose it is convenient to use the subharmonicity criterion of Theorem J7(iv). Thus, consider a closed disc, described by parameter values z E .1(0; 1), lying in the intersection of D with a complex line described parametrically by {A + Hz: z E q, and suppose that p is a complex polynomial in one variable such that -log d(A + Hz) ~ Re p(z) whenever z E o~(O; 1); what must be shown is that -log d(A + Hz) ~ Re p(z) whenever z E ~(O; 1). Now for any fixed value of z the inequality -log d(A + Hz) ~ Re p(z) can obviously be rewritten as d(A + Hz) ~ le-P(Z)I, and in view of Definition G3 that amounts to the condition that H(A + Hz; le-P(Z)!) ~ D if d = dD or to the condition that ~(A + Hz; le-p(z)IR) ~ D if d = bD,R' These two conditions in turn can quite evidently be rewritten in the common form A + Hz + We-p(z) E Dforall vectors Win an open subset E ~ en, where E = H(O; 1) if d = dD or E = ~(O; R) if d = bD,R' Therefore, if A + Hz + We-p(z) E D for all z E o~(O; 1), WEE, what must be shown is that A + Hz + We-p(z) E D for all z E ~(O; 1), WEE. For a fixed WEE, introduce the mapping F: [0, 1] x K(O; 1) --+ en defined by F(t, z) = A + Bz + Wte-P(:), noting that F is continuous and that F

130

Volume I

Function Theory

is holomorphic in L\(O; 1) for each fixed point in [0, 1]. The assumption above implies that F(t, z) E D_ if t E [0, 1], Z E aL\(O, 1), since clearly tW E E whenever t E [0, 1], while F(O x L\(O; 1)) £; D, since that is just the closed disc in D initially considered. If 1= {t E [0; 1]: F(t x L\(O; 1)) £; D}, it is evident from the continuity of F that I is an open subset of the unit interval, while it follows from the assumption that D is pseudoconvex in the sense of Hartogs that I is a closed subset of the unit interval. On the other hand, I is nonempty since 0 E I as observed above; hence the connectedness of the unit interval implies that 1= [0; 1]. That, of course, means that F([O; 1] x L\(O; 1)) £; D as desired. That condition (iv) implies condition (v) is quite obvious. It therefore remains only to show that condition (iii) or condition (v) implies condition (i), and again the same proof gives both implications. If -log dD is pseudoconvex and if K is a compact subset of D, then for any point A E KD necessarily -log dD(A) ~ C, where C = SUPZeK( -log dD(Z)), or equivalently dD(A) ~ e- c. But that means that KD is disjoint from the open neighborhood {Z ED: dD(A) < e- C } of the boundary aD in D, and as noted before that is all that is required to show that D is pseudoconvex. The same argument works for the distance function ~D.R for any fixed polyradius R, and the proof is thereby concluded. It is perhaps worth pointing out that the proof of the preceding theorem actually yields a slightly stronger result than stated, for the functions dD and ~D.R can be replaced by any function d of the form d(Z) = sup {t E ~ : Z + t WED for all vectors WEE}, where E is any open neighborhood of the origin in en with the property that tE £; E whenever t E [0, 1]. The particular shape of the set E, whether ball, polydisc, or something more general, really played no role in the proof. In practice, these more general distance functions d seem not as yet to have been very much used. Note that the condition that -log d is plurisubharmonic implies that lid = exp( -log d) is plurisubharmonic and the function lid also tends to + 00 upon approaching the boundary. So the theorem can be expressed in terms of properties of lid rather than of -log d if desired. It is also worth observing as a consequence of the preceding theorem that the condition that an open subset D £; en be pseudoconvex can be demonstrated by exhibiting a single plurisubharmonic function having suitable properties, such as the function -log dD , which tends to + 00 at each point of D. A variety of other functions will also suffice for this purpose. To be more precise, it is convenient to introduce the following useful auxiliary notion.

4. DEFINITION. A continuous function u: D -+ ~ in an open subset D £; en is an exhaustion function for D if for any real number r the set {Z E D: u(Z) ~ r} is a compact subset of D. In this definition the requirement that u be a continuous function is not unnatural, since the condition that the set {Z E D: u(Z) ~ r} be compact for any r implies that u is lower semicontinuous, while the functions of greatest interest here are also upper semicontinuous. It is, of course, possible to alter the preceding definition by requiring only that the closure of the set {Z E D: u(Z) ~ r} in C be

M Pseudoconvex Subsets of C"

131

a compact subset of D, or even just a subset of D. The latter condition merely amounts to requiring that the set {Z ED: u(Z) ~ r} be disjoint from an open neighborhood of iJD, and in both cases there is no longer any reason to require that the function u be continuous. The choice of definition is mostly a matter of taste, the essential results being more or less equivalent in any case. The form of the definition used here was chosen to parallel more closely the definitions of pseudoconvexity and holomorphic convexity. The relevance of this notion lies in the following simple consequence of the preceding theorem. An open subset D s;;; en is psuedoconvex if and only if it admits a continuous plurisubharmonic exhaustion function.

5. COROLLARY.

Proof. If D admits a continuous plurisubharmonic exhaustion function, then it is obvious from the definition that D is pseudoconvex. Conversely, if D is pseudoconvex, then by Theorem 3 the function -log dD is a continuous plurisubharmonic function in D. This function tends to + 00 at each point Z E iJD and so is itself a continuous exhaustion function if D is a bounded subset of en, but not necessarily otherwise. In the general case, though, note that the function v defined by v(Z) = IZll + ... + IZnl is a continuous plurisubharmonic function in D and hence that the function u = sup( -log dD , v) is also a continuous plurisubharmonic function in D. Now

{Z ED: u(Z)

~

r} = {Z ED: -log dD(Z)

~

r} n {Z ED: v(Z)

~

r}

and since the first set on the right-hand side of this equality is a closed subset of en contained in D while the second set is a closed bounded subset of en, their intersection is a compact subset of D. Therefore u is a continuous plurisubharmonic exhaustion function for D, and the proof is thereby concluded. It is perhaps worth remarking that the preceding corollary nicely exhibits the greater flexibility of plurisubharmonic than of holomorphic functions; a holomorphically convex subset D s;;; en does not admit an exhaustion function of the form If I where f is holomorphic in D. That is a well-known observation in the case n = 1, and is equally easy to see in the case n > 1. Indeed, if If I is an exhaustion function for a connected open subset D s;;; en where f E l!JD , n > 1, then f is nonzero outside a compact subset K s;;; D and thus Iffis holomorphic in D - K. It then follows from Hartogs's extension theorem, Theorem E6, that Iff is holomorphic in D. Since If I is an exhaustion function, Iff must attain its maximum modulus at some point of D and so must be constant by the maximum modulus theorem, and that is clearly an impossibility. Other properties of pseudoconvex domains, easily derived from Theorem 3, provide yet more examples illustrating the relative simplicity of pseudo convexity.

6. THEOREM.

If D. are pseudoconvex open subsets of en with D. s;;; D.+1' then their union

D= U.D. is also pseudoconvex.

Proof. If ZED,., so that the values dD,(Z) are well defined whenever v ~ J1., it is obvious that dD,(Z) ~ dD,+l (Z) and that dD(Z) = lim. dD,(Z). Consequently the functions -log dD,(Z) form a monotonically decreasing sequence of functions converg-

132

Volume I

Function Theory

ing to -log dD(Z). Since the sets Dv are assumed to be pseudoconvex, it follows from Theorem 3 that the functions -log dD,(Z) are plurisubharmonic, and by Theorem K5(g) the limit function -log dD(Z) is also plurisubharmonic. Indeed, -log dD(Z) is plurisubharmonic in any of the sets D" and hence in their union D, plurisubharmonicity being a local condition. Another application of Theorem 3 shows that D is then pseudoconvex, and that concludes the proof.

7. THEOREM.

The intersection of any two pseudoconvex sets in

en is also pseudoconvex.

Proof. If Dl and D2 are pseudoconvex sets in en and K £; Dl () D2 is a compact subset, then KD,nD2 £; lSD, () KD2 , since there are more plurisubharmonic functions in a smaller set. Thus, K D,nD2 is contained in a compact subset of Dl () D2 and so must itself be compact; that suffices to prove that Dl () D2 is pseudoconvex as desired.

8. THEOREM. If D is an open subset of en and if each point A E aD has an open neighborhood UA such that UA () D is pseudoconvex, then D is necessarily pseudoconvex. Proof. First suppose that D is a bounded open set in en satisfying the hypotheses of the theorem. It follows from Theorem 3 that for each point A E aD the function -log dUAnD is plurisubharmonic in the set UA () D. However, whenever Z E UA () D is sufficiently near A, it is obvious that -log dUAnD = -log dD(Z), and consequently -log dD is plurisubharmonic in D near A. But this is true for every point A E aD, and therefore -log dD is plurisubharmonic in an open neighborhood of aD in D. Since D is bounded, it follows that there is a compact subset K £; D such that -log dD is plurisubharmonic in D - K. Now choose a point A ¢ K and introduce the continuous plurisubharmonic function v in en defined by v(Z) = rLi IZi - ail, where r > 0 is sufficiently large that v(Z) ~ -log dD(Z) for all points Z in an open neighborhood of K. There obviously exists such a constant r, since v(Z) # 0 whenever Z E K and K is compact. The function u = sup (v, -log dD) is then plurisubharmonic in D - K, since both v and -log dD are, and coincides with v in an open neighborhood of K so is also plurisubharmonic there. Thus, u is a continuous plurisubharmonic function in D. On the other hand, u tends to + 00 upon approaching any point of aD, and since D is bounded, that implies that u is an exhaustion function for D. It then follows from Corollary 5 that D is pseudo convex as desired. Next if D is an arbitrary open set in en satisfying the hypotheses of the theorem, it is apparent that the intersection D () B(O; v) for any integer v ~ 0 is a bounded open set in en also satisfying the hypotheses of the theorem, in view of Theorem 7 and the observation that B(O; v) is a domain of holomorphy and hence is pseudoconvex. It then follows from the first part of the proof of this theorem that D () B(O; v) is pseudoconvex; but since D () B(O; v) £; D () B(O; v + 1) and D = UvD () B(O; v), it further follows from Theorem 6 that D is pseudoconvex, and that suffices to conclude the proof.

The preceding theorem shows that the pseudoconvexity of an open subset

D

£;

en is a local property of aD. If D is pseudoconvex, then Theorem 7 shows that

each point A

E

aD has arbitrarily small open neighborhoods UA such that UA

()

D

M Pseudoconvex Subsets of en

133

is pseudo convex, when UA are arbitrarily small balls centered at A, for example, while Theorem 8 shows conversely that this local pseudoconvexity of D implies that Dis pseudoconvex. It is interesting to note, as an immediate corollary of Theorems 3 and 8, that if -log d Dis plurisubharmonic in an open neighborhood of aD in D, then it is necessarily plurisubharmonic throughout D. The same assertion of course holds for the function -log ~D.R for any polyradius R. As already observed, any holomorphically convex set in en is pseudoconvex. After the inverse implication is established, and it is thus demonstrated that holomorphic convexity and pseudoconvexity are equivalent notions for open sets in en, the preceding results extend immediately to hold for domains ofholomorphy. These results are very useful and not at all easy to demonstrate directly. The analogue for domains of holomorphy of Theorem 6, the assertion that an increasing union of domains ofholomorphy in en is again a domain ofholomorphy, was first established by Behnke and Stein. The problem of demonstrating that holomorphic convexity is a local property-that is, of demonstrating the analogue of Theorem 8 for domains ofholomorphy-is usually referred to as the Levi problem. This problem was solved in e 2 by Oka, and his solution was extended to en by Bremermann and Norguet. In one special case this result can be obtained quite simply, by comparing Theorems D11 and GlO, and the following. 9. THEOREM. A tube domain in en with base B is pseudoconvex precisely when B is a convex subset of ~n in the usual sense. Proof. For a tube domain D = B + mn £; en, it is evident that dD(Z) depends only on the real part of Z and hence can be viewed as a function in B, and it then follows from Theorem K13 that -log dD(Z) is a plurisubharmonic function of ZED precisely when -log dD(X) is a convex function of X E B. So from Theorem 3 and one of the known characterizations of convex sets, D is pseudoconvex precisely when B is convex. That suffices to conclude the proof.

With some of the basic elementary properties of pseudoconvex sets having been demonstrated, it is perhaps now worth returning to the definition of pseudoconvexity in order to examine other notions of convexity arising from the choice of families of plurisubharmonic functions other than the family of continuous plurisubharmonic functions. Actually all of these notions coincide, as will next be seen. First, to begin with the family of all plurisubharmonic functions, note the following result. 10. THEOREM. If D is a pseudoconvex open subset of en and K is a compact subset of D with plurisubharmonically convex hull K D, then for any plurisubharmonic function u in D and any point A E K D, necessarily u(A) ~ sUPZeKu(Z). Proof. The conclusion, of course, holds for any continuous plurisubharmonic function u by the very definition of pseudoconvexity; the point is to show that it holds for an arbitrary not necessarily continuous plurisubharmonic function u. Suppose, to the contrary, that there exist a plurisubharmonic function u and a point A E KD such that u(A) > sUPZeK u(Z). and choose a constant m with u(A) > m >

134

Volume I

Function Theory

suPZ EK u(Z). Since U is upper semicontinuous, the set U = {Z ED: u(Z) < m} is an open neighborhood of K in D. Choose a continuous plurisubharmonic exhaustion function v for the set D, noting that the existence of such a function is guaranteed by Corollary 5, since D is pseudoconvex by hypothesis. By adding a suitable real constant to v, it can evidently be arranged that v(Z) ~ 0 whenever Z E AuK. Then for any fixed value b > 0, the ~et V = {Z ED: v(Z) < b} is an open neighborhood of AuK in D, and its closure V ~ {Z ED: v(Z) ~ b} is a compact subset of D. Now consider the function u, associated to U as in Definition K9, where e is chosen sufficiently small that A(Z; e) ~ U ~ D whenever Z E K and that A(Z; e) ~ D whenever Z E V. Recall from Theorem K 11 that u, is a COO plurisubharmonic function in the open subset D, = {Z ED: bD(Z) > e}, and that u,(Z) ~ u(Z) whenever ZED,; here e has been chosen sufficiently small that AuK ~ V ~ D,. Note that whenever Z E K and WE A(O; 1), then Z + eW E A(Z; e) ~ U so that u(Z + eW) < m, and hence from equation K(3) it follows that u,(Z) = fWEd(O; 1) u(Z +eW)O'(W) dV(W) < fWEMO;l)mO'(W) dV(W) = m. Therefore, altogether, u,(A) ~ u(A) > m > u,(Z) ~ u(Z)

whenever

Z

E

K

(1)

Finally, choose a real constant r sufficiently large that u,(Z) < rb

+ m whenever Z E oV

(2)

as is evidently possible since oV is a compact subset of D" and introduce the functions v+ and w in D defined by v+(Z) = sup(v(Z), 0) for any ZED and w(Z)

= {SUP(U,(Z), rv+(Z) + m) rv+(Z)

+m

if Z E V if ZED-V

(3)

Note that v+ is a continuous plurisubharmonic function in D, while w is a continuous plurisubharmonic function in the two open subsets V and D - V of D. Moreover, if Z E 0 V, then it follows from the definition ofthe set V that v(Z) = b > 0, and hence from (2) that u,(Z) < rb + m = rv+(Z) + m. But then in view of the definition (3) of the function w, it is evident that w coincides with rv+ + m in an open neighborhood of oV in D, and therefore w is actually a continuous plurisubharmonic function in D. Now if Z E K ~ V, then from (1) it follows that u,(Z) < m ~ rv+(Z) + m and therefore w(Z) = rv+(Z) + m = m, since by construction v(Z) ~ 0 whenever Z E K. But on the other hand, for the point A E V it is also the case that v(A) ~ 0 by construction, so it follows from (1) that u,(A) > m = rv+(A) + m and consequently w(A) = u,(A) > m = SUPZEK w(Z). That contradicts the assumption that A E KD and thereby concludes the proof. This theorem shows that in a pseudoconvex open set D ~ en the plurisubharmonicially convex hull of any compact subset K ~ D can be defined either by using only continuous plurisubharmonic functions as in Definition 1 or by using arbitrary

M Pseudoconvex Subsets of

en

135

plurisubharmonic functions. Consequently, with the latter definition the plurisubharmonically convex hull is still necessarily a closed subset of D-indeed, a compact subset of D-even though the separate sets {A ED: u(A) ~ sUPzeKu(Z)} need not be closed when u is not continuous. Although this was established only for pseudoconvex open sets in en, it is true without the explicit assumption of pseudoconvexity in the following sense.

11. THEOREM.

For an open subset D

~

en the following conditions are equivalent:

(i) D is pseudoconvex. (ii) Whenever K ~ D is compact, then the subset { A ED: u(A)

~ sup u(Z) for all plurisubharmonic functions u in D} ZeK

is also compact.

(iii) Whenever K { A ED: u(A)

~

D is compact, then the subset

~ sup u(Z) for all plurisubharmonic functions u in D} ZeK

is disjoint from an open neighborhood of oD.

Proof. The subset of D described in the statement of condition (ii) is of course contained in KD , and if Dis pseudoconvex, it follows from Theorem 10 that this subset actually coincides with KD , which is compact; thus, condition (i) implies condition (ii). It is trivial that condition (ii) implies condition (iii). It remains only to show that condition (iii) implies condition (i), and for that purpose it is convenient to recall from Theorem 3 that D is pseudoconvex precisely when D is pseudoconvex in the sense of Hartogs. Suppose therefore that D satisfies condition (iii), and consider a continuous mapping F: [0, 1] x ;;\(0; 1) -+ en such that F is holomorphic in .::\(0; 1) for each fixed point of [0, 1] and that F([O, 1) x ;;\(0, 1» u F(1 x 0.::\(0; 1» ~ D. The image F([O, 1] x 0.::\(0; 1» = K is thus a compact subset of D, so by condition (iii) the set L = {A E D: u(A) ~ sUPZeKu(Z) for all plurisubharmonic functions u in D} is disjoint from an open neighborhood of oD, or equivalently its closure is a subset L ~ D. For any fixed point t E [0, 1), any constant e > 0, and any plurisubharmonic function u in D, note that since F(t x 0.::\(0; 1» ~ K and u is upper semicontinuous in D, then u(A) < SUPZeK u(Z) + e for all points A in an open neighborhood of F(t x 0.::\(0; 1» in D, and consequently u(F(t, z» < SUPZeK u(Z) + e for all points z in an open neighborhood of 0.::\(0; 1) in .::\(0; 1). But u(F(t, z» is actually a subharmonic function of z E .::\(0; 1), since F(t, z) is a holomorphic function of z, so it follows from the maximum theorem for subharmonic functions that u(F(t, z» < sUPZeKU(Z) + e for all points z E .::\(0; 1). This is true for any value e > 0, and therefore u(F(t, z» ~ sUPZeK u(Z) for all points z E .::\(0; 1). But the latter inequality is true for any plurisubharmonic function u, and therefore

136

Volume I

Function Theory

F(t, z) E L for all points Z E A(O; 1). Thus, altogether F([O; 1) x ~(O; 1» £ L, and since F is continuous while L £ D, it follows that F([O, 1] x ~(O; 1» £ L £ D. Therefore, D is pseudoconvex in the sense of Hartogs, and the proof is thereby completed.

Then for the consideration of the corresponding notions of convexity involving various naturally occurring subfamilies of the family of continuous plurisubharmonic functions, what is needed to settle matters is just the following result.

An open subset D £ en is pseudoconvex if and only if it admits a c n strictly plurisubharmonic exhaustion function.

12. THEOREM.

Proof. If D admits such an exhaustion function, then D is pseudoconvex as a consequence of Corollary 5. For the other implication, which is of course the main point of the theorem, suppose that D is pseudoconvex and choose a continuous plurisubharmonic exhaustion function u for D, the existence of such a function being guaranteed by Corollary 5. For any integer v, the set Dv = {Z ED: u(Z) < v} is then an open subset of D, and its closure Dv is a compact subset of D. Now consider the functions u. associated to u as in Definition K9, recalling from Theorem Kll that u. is a COO plurisubharmonic function in the open subset D. = {Z ED: t5D(Z) > e} and that u.(Z) ~ u(Z) whenever ZED•. Since u is continuous, the functions u. converge uniformly to u on any compact subset of D as e tends to zero by Lemma KI0. Thus, if ev is chosen sufficiently small, then Dv £ D., and u(Z) ~ u.,(Z) < u(Z) + 1 whenever Z E Dv. The expression IIZII 2 is easily seen to be a COO strictly plurisubharmonic function of Z E en, since a trivial calculation shows that its Levi form is LII·1I 2(Z; A) = IIAII2. Therefore, the function U v defined by uv(Z) = u. (Z) + t5v11Z112 is clearly a Coo strictly plurisubharmonic function in an open neighborhood of Dv for any t5v > 0, and if t5v is chosen sufficiently small, it can be arranged that u(Z) < uv(Z) < u(Z) + 1 whenever Z E Dv. Next choose a Coo monotonically increasing function f/J: IR -+ IR such that f/J(t) = 0 whenever t < -!, f/J(t) = 1 whenever t > 2, and f/J'(t) > 0, f/J"(t) > 0 whenever -! < t < !. In terms of this auxiliary function f/J and the previously constructed functions u v , introduce the further set of functions Vv defined by

vv(Z) =

{i(Uv(Z) + 2 -

v)

if Z E Dv if ZED - Dv

(4)

Note that u is a Coo function in an open neighborhood of Dv , and that if Z E 8Dv , then uv(Z) + 2 - v > u(Z) + 2 - v = 2; thus, uv(Z) + 2 - v > 2 for all points Z in an open neighborhood of the compact set 8Dv, so that f/J(uv(Z) + 2 - v) = 1 for all such points Z. It follows from this that Vv is actually a Coo function in the entire set D. Next note that whenever Z E Dv- s , then uv(Z) + 2 - v < u(Z) + 3 - v ~ -2 and f/J(uv(Z) + 2 - v) = 0; thus,

vv(Z)

=0

if Z

E

Dv - s

Further note that whenever Z

(5) E

Dv- 2 , then uv(Z) + 2 -

v < u(Z)

+3-

v ~ 1, and

M Pseudoconvex Subsets of C'

137

therefore uv(Z) + 2 - v < ~ in an open neighborhood of Dv- 2 • Since ¢ is a monotonically increasing convex function in ( - 00, ~), it follows from Theorem K5(d) that Vv is a plurisubharmonic function in an open neighborhood of Dv - 2 ' Finally note that whenever Z E Dv- 2 - Dv- 3 , then 1 ~ uv(Z) + 2 - v > u(Z) + 2 - v ~ -1, and therefore ~ > uv(Z) + 2 - v > -~ in an open neighborhood of the compact set D'-2 - D.- 3 • Since fil(t) > 0, ¢"(t) > 0 whenever -~ < t < ~, and the Levi form of the function v. can be written in D. as

it follows that v. is actually a strictly plurisubharmonic function in an open neighborhood of D.-2 - D.- 3 . From these properties it is easy to see that it is possible to choose real numbers c. ~ 1 sufficiently large that w. = C 1 V 1 + ... + c.v., which is a COO function in D, is plurisubharmonic in D'-2 for all integers v ~ 1. Indeed, to proceed inductively on v, set C 1 = 1 and note that W1 = V 1 is plurisubharmonic in an open neighborhood of D-1 as pointed out above. If Wv is plurisubharmonic in an open neighborhood of Dv - 2 for some integer v ~ 1, note that Vv+1 is plurisubharmonic in an open neighborhood of DV - 1 and indeed is strictly plurisubharmonic in an open neighborhood of the compact set DV - 1 - Dv - 2 ' Since Wv + 1 = Wv + Cv +1 Vv+1' it is then possible as in Theorem L2 to choose C v+1 ~ 1 sufficiently large that Wv+ 1 is plurisubharmonic in an open neighborhood of the compact set Dv - 1 - Dv - 2 , and then W.+1 will be plurisubharmonic in an open neighborhood of D'-1 as desired. Note as a result of (5) that V.+ 1 vanishes identically in D). whenever v ~ 2 + 4, and consequently W.+ 1 = W. + C.+ 1 V = w. identically in D). whenever v ~ 2 + 4. The limit w = lim .....oo w. is therefore a well-defined COO plurisubharmonic function in D, since the sequence w. is eventually stationary on any subset D). s; D. Moreover, note as a result of the definition (4) that whenever Z ¢ D)., then v.(Z) = 1 for all values v ~ 2, and therefore w(Z) ~ w).(Z) ~ C 1 + ... + c). ~ 2. The function w is therefore a COO plurisubharmonic exhaustion function for D. To obtain a strictly plurisubharmonic such exhaustion function, it is only necessary to replace w by the function associating to each point ZED the value w(Z) + bllZI1 2 for any b > 1; that suffices to conclude the proof. Example. That holomorphically convex sets are the same as domains of holomorphy might suggest that pseudoconvex sets are possibly the same as the natural domains of existence of plurisubharmonic functions. It follows from Corollary 5, for instance, that in every pseudoconvex open subset D s; en there is a plurisubharmonic function that tends to + 00 upon approaching aD and hence that cannot be extended to a plurisubharmonic function in any properly larger set. On the other hand, it is not the case that any plurisubharmonic function in an open subset D s; en necessarily extends to a plurisubharmonic function in the smallest pseudoconvex set containing D. The following simple example of that is due to H. Bremermann. Consider the tube domain D S; e 2 with base B = B1 U B 2, where BI = {(Xl' X2) E ~2: 2 < SUP(IXII, Ix 2 1) < 4} and B2 = {(Xl' X2) E ~2: 0 < Xl ~ 2, IX21 < I}, as sketched in Figure 9. The base B is not convex, so by Theorem 9 the

138

Volume I

Function Theory

Figure 9

set D is not pseudoconvex. If every plurisubharmonic function in D extends to a pseudoconvex set E ::::l D, then since D is preserved by arbitrary translations in the imaginary direction, it is evident that E must have the same property, so that E must also be a tube domain. Indeed, clearly E must be the tube domain with base B the convex hull of B, the set B = {(Xl' X 2 ) E ~2 : sup(lxd, Ix 2 1) < 4}. Now the function u in B defined by if if

(Xl' X 2 )

E Bl

(X l ,X2)EB2

is of class C 2 and is convex, since ipujaxi = 6(2 - xd ~ 0 in Bl and a2 ujaxi = 0 if Xl = 2. Thus by Theorem K13, when viewed as a function in D depending only on the variables (Xl' x 2 ), the function u is plurisubharmonic in D. It is clear that this function u cannot be extended to a plurisubharmonic function in E, for since u is zero in the tube over Bl but takes positive values at points in the tube over B2 , the extended function would attain a positive maximum value at an interior point of E, contradicting the maximum theorem for plurisubharmonic functions.

N Pseudoconvex Riemann Domains

The notion of pseudoconvexity can easily be extended to Riemann domains, and the results about pseudoconvex open subsets of en obtained in the preceding section can be shown also generally to hold for Riemann domains. 1. DEFINITION. A Riemann domain M is pseudoconvex if for any compact subset K !;; M the set KM = {A E M: u(A) ~ sUPzeKu(Z) whenever u is a continuous plurisubharmonic function on M} is also compact. The set KM is the plurisubbarmonically convex bull of K inM.

The distance functions dM and ~M.R can be introduced for Riemann domains as in Definition H2, paralleling the corresponding definitions for open subsets of en, and can be used to characterize pseudoconvex Riemann domains in the same way that they were used to characterize pseudoconvex open subsets of en. The definition of an exhaustion function can be extended without change to Riemann domains, and can be shown to play the same role for Riemann domains as for open subsets of en in the discussion of pseudoconvexity. On the other hand, the notion of pseudoconvexity in the sense of Hartogs requires some slight modification in its extension to Riemann domains, for the condition that an open subset D !;; en be pseudoconvex in the sense of Hartogs as in Definition M2 involved properties of mappings into the canonical pseudoconvex set en containing D, while Riemann domains cannot always be viewed as subsets of some canonical larger pseudoconvex Riemann domains. Indeed, it is perhaps worth mentioning here, without going into the details at present, though, that there exist noncom pact Riemann domains that are maximal in the sense that they cannot be viewed as proper subsets of any other Riemann domains. 2. DEFINITION. A Riemann domain M with projection P: M -+ en is pseudoconvex in tbe sense of Hartogs if whenever-.!: [0, 1] x [\(0; 1) -+ en is a continuous mapping from the compact subset [0, 1] x A(O; 1) !;; ~ x e into en such that F is holomorphic in A(O; 1) for each fixed point of [0, 1] and G: ([0, 1) x [\(0; 1» u (1 x oA(O; 1» -+ M

140

Volume I

Function Theory

is a continuous mapping such that P mapping G: [0, 1] x [\(0; 1) -+ M.

0

G = F, then G can be extended to a continuous

It should be noted that the continuity of the extension Gimplies that po G= F and that Gis uniquely determined; the mapping Gis of course also holomorphic in .1(0; 1) for each fixed point of [0, 1]. If M is an open subset of en and P: M -+ en is the inclusion mapping, then the preceding definition reduces immediately to Definition M2. In general, the preceding definition modifies Definition M2 merely by replacing the condition that the values of F lie in D by the condition that F lifts to a mapping into M.

3. THEOREM.

For a Riemann domain M, the following conditions are equivalent:

M is pseudoconvex. M is pseudoconvex in the sense of H artogs. The function -log d M is plurisubharmonic on M. For any polyradius R the function -log (jM,R is plurisubharmonic on M. There is a polyradius R such that the function -log (jM,R is plurisubharmonic on M. (vi) M admits a continuous plurisubharmonic exhaustion function. (vii) M admits a C''' strictly plurisubharmonic exhaustion function. (i) (ii) (iii) (iv) (v)

Proof. The demonstration that condition (i) implies condition (ii) is essentially the same as the demonstration of the corresponding assertion in the proof of Theorem M3. If M is a pseudo convex Riemann domain with projection P: M -+ en and if F: [0, 1] x [\(0; 1) -+ en and

G: ([0, 1) x [\(0; 1)) u (1 x aA(O; 1)) -+ M are mappings as in Definition 2 with po G = F, then the image K = G([O, 1] x aL\(O; 1)) is a compact subset of M and hence has a compact plurisubharmonically convex hull KM . For any fixed point Z E [\(0; 1) the set of points G(t, z) for t E [0,1) must therefore have at least one limit point G(I, z) E M as t approaches one; but since P is then a homeomorphism from an open neighborhood of G(I, z) in M onto an open neighborhood of P(G(I, z)) = F(I, z) in en, it follows immediately that G(t, z) extended to the value t = 1 by G(I, z) is actually a lifting of the full path {F(t, z): 0 ;£ t ;£ I}. That provides the desired extension of the mapping G and shows that M is pseudoconvex in the sense of Hartogs as desired. The demonstration that condition (ii) implies conditions (iii) and (iv) is also very much like the demonstration of the corresponding assertion in the proof of Theorem M3. Suppose that M is a Riemann domain that is assumed to be pseudoconvex in the sense of Hartogs, let P: M -+ en be its projection, and as in the proof of Theorem M3, let d denote either dM or (jM,R' Since it is easily seen that d is continuous, in order to show that -log d is plurisubharmonic, it is sufficient to show that its restriction to any complex line through any point of Mis subharmonic, the lines being those defined in terms of the coordinates induced on M by the projection P: M -+ ICn; again it is convenient t9 use the subharmonicity criterion of

N Pseudoconvex Riemann Domains

141

Theorem J7(iv) for this purpose. Thus, suppose that H is a hoi om orphic mapping from an open neighborhood of the closed unit disc L\(O; 1) ~ e into M such that P(H(z» = A + Bz for some A, BEen, and suppose that p is a complex polynomial in one variable such that -log d(H(z» ~ Re p(z) whenever z E oA(O; I)-that is, such that d(H(z» ~ Ie-p(Z) I whenever z E oA(O; 1). What must be shown is that -log d(H(z» ~ Re p(z) whenever z E A(O; I)-that is, that d(H(z» ~ le-P(Z)I whenever z E A(O; 1). Recall from Definition H2 that d(H(z» is the largest real number for which there exists an open neighborhood ofthe point H(z) in M that is mapped homeomorphically by the projection P to the open neighborhood P(H(z» + rE of the point P(H(z» in en, where E = B(O; 1) if d = dM or E = A(O; R) if d = c5M ,R' For any point WEE, introduce the mapping Fw: [0, 1] x L\(O; 1) -+ en defined by Fw(t, z) = P(H(z» + te-p(z)w. Note that Fw(O, z) = P(H(z», and therefore that the mapping Gw: 0 x L\(O; 1) -+ M defined by Gw(O, z) = H(z) has the property that P(Gw(O, z» = Fw(O, z). Now for any fixed point z E L\(O; 1), it is clearly possible to extend Gw to a continuous mapping Gw: [0, c5z ) x z -+ M such that P(Gw(t, z» = Fw(t, z). Since L\(O; 1) is compact, there is therefore an extension of Gw to a mapping Gw: [0, (j) x L\(O; 1) -+ M for some value (j > 0, such that P(Gw(t, z» = Fw(t, z); in particular, let c5 be the largest value in [0, 1] for which there exists such an extension, and note that necessarily c5 = 1. Indeed, since d(H(z» ~ le-P(Z)I whenever z E oA(O; 1) and since Fw(t, z) E P(H(z» + le-p(z)IE whenever t E [0, 1], it follows that the mapping Gw can be extended to a continuous mapping

Gw : ([0, (5) x L\(O; 1» u ([0, 1] x oA(O; 1» -+ M such that P(Gw(t, z» = Fw(t; z). But since M is assumed to be pseudoconvex in the sense of Hartogs, it follows that Gw can be extended to a continuous mapping

Gw: ([0, c5] x L\(O; 1» u ([0, 1] x oA(O; 1» -+ M such that P(Gw(t, z» = Fw(t, z), and if (j < 1 it is easy to see using continuity alone that there is a further extension of Gw to a continuous mapping Gw : [0, (j') x L\(O; 1) -+ M covering Fw for some value (j' > (j, contradicting the maximality of (j. Thus, there must exist a continuous mapping Gw: [0, 1] x L\(O; 1) -+ M such that Gw(O; z) = H(z) and P(Gw(t, z» = P(H(z» + te-p(z)W for any element WEE; but that is evidently enough to show that d(H(z» ~ le-P(Z)I whenever z E A(O; 1), as desired. That condition (iv) implies condition (v) is obvious. The proof that condition (iii) or (v) implies condition (vi) is necessarily somewhat more difficult than the proof of the corresponding implication for open subsets of en in Corollary M5, because it is necessary to take into account the possibility that the Riemann domain is not finitely sheeted. The proof is nonetheless considerably easier that the proof of the analogous results for holomorphic rather than for plurisubharmonic functions, as will be evident upon comparing the argument to follow with that found in section I; that reflects yet again the greater abundance and flexibility of plurisubharmonic functions. Let M be a connected Riemann domain with projection P: M -+ en, and suppose that -log d is plurisubharmonic

142

Volume I

Function Theory

on M where d stands for either dM or bM,R' The first step in the proof is the construction of an auxiliary continuous exhaustion function for M. Choose a fixed but arbitrary base point B E M, and note that since M is connected, any other point Z E M can be joined to M by a continuously differentiable path. The length of that path can be defined by projecting it locally to en by the projection P: M -+ en and calculating the length piecewise by the usual differential geometric formulas in en = 1R2n. Let w(Z) denote the infimum ofthe lengths of all such paths from B to Z; it is clearly a continuous exhaustion function w for M. The function w need not be differentiable, for instance, at those points Z that can be joined to B by several distinct paths of minimal length. It is nonetheless obvious, though, that whenever Zl and Z2 are points of M that are near enough to lie in an open subset of M mapped homeomorphically into en by the projection P, then (1)

The next step consists in modifying this function w to secure a set of COO strictly plurisubharmonic functions that are in some sense approximate exhaustion functions. For any positive integer v, introduce the subset M.= {Z EM: -log d(Z) < v} = {Z EM: d(Z) > l/v}, and note that there is a constant e > 0 such that each point Z E M. has an open neighborhood that is mapped homeomorphically onto an open neighborhood of the closed polydisc l\(P(Z); e) by the projection P: M -+ en. Indeed, since d(Z) > l/v, there exists an open neighborhood of Z mapped homeomorphically onto B(P(Z); l/v) if d = dM or onto A(P(Z); R/v) if d = bM,R' and it is only necessary to choose e sufficiently small that A(O; e) c: A(O; l/v) or A(O; e) c: A(O; R/v). Use the projection mapping P to identify an open neighborhood of Z E M with A(P(Z); e); it is then possible to introduce the function w. associated to w as in Definition K9, in terms of the coordinates in en. The function w. is thus a welldefined COO function on M., and it is easy to see that it is an approximate exhaustion function for M in the sense that for any real number c the closure of the set {Z EM: w.(Z) < c} is a compact subset of M. The function w. is of course not really an exhaustion for M, since it is not defined everywhere on M, nor is it an exhaustion function for M., since it is only asserted that the closure of the set {Z E M. : w.(Z) < c} is a compact subset of M rather than of M •. To see this, note that for any real number c the set X = {Z EM: w(Z) < c} has a compact closure in M, and therefore the set X. = Uzex l\(Z; e) is a compact subset of M, where l\(Z; e) is the closure of the open neighborhood of Z in M mapped homeomorphically to A(P(Z); e) under the projection P. If Z ¢ X., then the function w is at least equal to c throughout the open neighborhood A(Z; e), and it follows from the defining formula K(3) that w.(Z) ~ c; thus, {Z E M.: w.(Z) < c} must lie in the compact subset X. ~ M and hence must have compact closure. Next to calculate the Levi form of this function w. in terms of the natural coordinates on M induced by the projection P: M -+ en, identifying an open neighborhood ofthe point Z E M with the polydisc A(Z; e) ~ en and recalling the defining equation K(3) in which u(Z) = u(z 1)" . u(zn) imply that

N Pseudoconvex Riemann Domains

=

~

=

_!~

W(S)U(Sl - Zl) ... U(S" - Z")B- 2 " dV(S)

aZj aZk JSeA(Z;.) B aZj

a

1 -a = -B

=

r

Zj

_! lim B 6-+0

B

r

W(S)U(Sl -

B

JseA(Z;.)

f f

143

B

Zl)"'Ur(Sk - Zk) ... u(s" - Z")B- 2 " dV(S) B B

w(Z

+ BT)u(td'"

w(Z

+ ~Ej + BT) -

Ur(t k)··· u(t,,) dV(T)

TeA(O;l)

~

TeA(O;l)

w(Z

+ BT) u(t 1)' .. U,(t k)· .. U(t,,) dV(T)

where Ur = au/at and Ej = (0, ... , 1, ... ,0) is the unit vector with entry 1 in the jth place. From (1), note also that

and consequently that la 2 w.(Z)/azj Ozkl ~ CB- 1 , where C = IJeuf(t) dV(T)I. Altogether, therefore,

~ CB- 1

L Iajil I ~ CB- n211A112 1

k

j,k

for any point Z on Mv by

E Mv

and any vector A

E

en. Now introduce the function Vv defined

noting that ILvv(Z; A)I = ILw.(Z; A) + 2CB- 1 n211A11 21~ CB- 1 n211A1I2 and hence that Vv is a COO strictly plurisubharmonic function on M. Clearly this function too has the property that for any real number c, the closure of the set {Z E M: vv(Z) < c} is a compact subset of M. The final step consists in combining the plurisubharmonic functions -log d and Vv appropriately so as to obtain a continuous plurisubharmonic exhaustion function for M. For any real constants Cv the sets Kv = {Z EM: -log d(Z) < v and Vv+2(Z) < cv} and Lv = {Z E ~: -log d(Z) < v + t and V.+2(Z) < c. + I} have compact closures in M and Kv s; Lv; and beginning with C 1 = 1 and choosing

144

Volume I

Function Theory

sufficiently large constants Cy inductively on v yield sets for which Ly ~ K + 1 and Uy Ky = Uy Ly = M. For such constants Cyit is then possible to show by induction on v that there exists a sequence of functions Uy such that Uy is a continuous plurisubharmonic function on M y+1 , that uy(Z) > Jl. whenever Z E Ly - L" for Jl. < v, and that uy(Z) = U y- 1 (Z) whenever Z E K y- 1 . The function U defined by u(Z) = limy uy(Z) is then clearly a continuous plurisubharmonic exhaustion function on M as desired. The induction can be begun by setting u 1 = V2; then if the functions U 1 , ••• , U Y - 1 have been constructed for some index v ~ 2, it is only necessary to construct a continuous plurisubharmonic function U y in M Y+ 1 such that uy(Z) > v - I whenever Z E Ly - L y- 1 , that uy(Z) > v - 2 whenever Z E Ly - L y - 2 , and that uJZ) = U y- 1 (Z) whenever Z E Ky- 1 • For this purpose, note that the function uy defined on M Y+ 1 by Y

Uy(Z) = sup( -log d(Z) - v

+ 1, vy+ 1 (Z) -

cy-d

is a continuous plurisubharmonic function on M y+ 1 , and that K y- 1 = {Z EM: uy(Z) < O}. The compact subsets L y - L y- 1 and oL y_1 are disjoint from Ky- 1 ; hence, it is possible to choose a real constant by sufficiently large that byuy(Z) > v - I whenever Z E Ly - L y- 1 and that byuy(Z) > U y- 1 (Z) whenever Z is in an open neighborhood of oL y - 1 • The function U y defined in M Y +1 by

U

y

(Z) = {SUP(U Y _ 1 (Z), byuy(Z» byuy(Z)

if Z if Z

E

L y-

E

MY +1

1

-

L y-

1

is then easily seen to have the desired properties, thus concluding this part of the proof of the theorem. The proof that condition (vi) implies condition (vii) is exactly the same as the proof of the corresponding implication for open subsets of e" in Theorem MI2. From a continuous plurisubharmonic exhaustion function u on M, the COO functions U ycan be introduced on M as the proof of the preceding implication just above, and the remainder of the argument proceeds formally the same on a Riemann domain or in an open subset of e". Finally it is quite obvious that condition (vi) or (vii) implies condition (i), and that suffices to conclude the proof of the entire theorem. The preceding theorem provides a very direct extension of some properties of pseudoconvex open subsets of e" to the corresponding properties of pseudoconvex Riemann domains. The statements are the same in both cases, although some of the proofs are rather more complicated in the case of Riemann domains. On the other hand, there are difficulties even in extending the statements of some other properties of pseudoconvex open subsets of e" to the case of Riemann domains; for instance, it is not always possible to make sense of the intersection of two Riemann domains, nor is it always possible to speak of the boundary of a Riemann domain, since as mentioned earlier there are maximal noncompact Riemann domains. These difficulties can be avoided quite naturally by limiting the consideration to subsets of a fixed Riemann domain, reflecting the fact that the open subsets of e" are after all

N Pseudoconvex Riemann Domains

145

subsets of the same fixed domain en. Any open subset of a Riemann domain inherits the natural structure of a Riemann domain by restricting the projection mapping, so it is possible to speak of any subset of a Riemann domain as being a pseudoconvex set. From these observations, Theorems M6, M7, and M8 extend to Riemann domains as follows. 4. THEOREM. If a Riemann domain M can be written as the union of open subsets M. £;; M where M. £;; M.+1and each subset M. is pseudoconvex, then M is itselfpseudoconvex.

Proof. As in the proof of Theorem M6, the function -log dM is the limit of the monotonically decreasing sequence offunctions -log dM ,. If M. are pseudoconvex, then the functions -log dM , are plurisubharmonic by Theorem 3, the limit function -log dM is then also plurisubharmonic by Theorem K5 (g) and another application of Theorem 3 shows that Mis pseudoconvex as desired, thus concluding the proof. 5. THEOREM. If MI and M2 are pseudoconvex open subsets of a Riemann domain M, then MIn M 2 is also pseudoconvex.

Proof. As in the proof of Theorem M7, for any compact subset K £;; MI n M 2 , clearly KMlnM2 £;; KMI n K M2 , from which the desired result is an immediate consequence. 6. THEOREM.

If M is an open subset of a pseudoconvex Riemann domain N and if each point

A E aM has an open neighborhood UA in N such that UA n Mis pseudoconvex, then M is itself pseudoconvex.

Proof. First suppose that the point set closure of M in N is a compact subset M £;; N. As in the proof of Theorem M8, for each point A E aM it is clear that -log dM(Z) = -log duAnM(Z) whenever Z E JtA n M for a sufficiently small open subneighborhood JtA £;; UA of A. Since UA n M is pseudoconvex by hypothesis, it follows from Theorem 3 that -log dUAnM is plurisubharmonic inYA n M, and consequently -log dM is plurisubharmonic in UA (JtA n M). Since M is compact, it is evident that K = M - UA (VA n M) is also compact. Note that the function v on N defined by v(Z) = 1 + IPI (z)1 + ... + IpiZ) I is plurisubharmonic on N and bounded away from zero on the compact subset KeN, when P(Z) = (PI (Z), ... , Pn(Z)), For a positive constant r sufficiently large that rv(Z) > -log dM(Z) whenever Z E K, it follows that the function u = sup( -log dM , rv) is a continuous plurisubharmonic function on M. Since u tends to + 00 upon approaching the boundary of M, the function u is a continuous plurisubharmonic exhaustion function for M, and consequently M is pseudoconvex. Next for the general case ofthe theorem, since N is pseudoconvex by assumption, it follows from Theorem 3 that N admits a continuous plurisubharmonic exhaustion function u. If M is an arbitrary open subset of N satisfying the hypotheses of the present theorem, it is apparent that for any integer v the subset M. = {Z EM: u(Z) < v} also satisfies the hypothesis of this theorem and has a compact closure in M. It then follows from the first part of the present proof that M. is

146

Volume I

Function Theory

pseudoconvex, and since M. s;;; M.+l and U.M. = M, it follows from Theorem 4 that M is pseudoconvex. That suffices to conclude the proof. As in the proofs of Theorems MIO and Mll, it follows that for Riemann domains just as for open subsets of en, pseudoconvexity can be defined equivalently in terms of arbitrary rather than of continuous plurisubharmonic functions. It was already demonstrated in Theorem 3 that pseudoconvexity can also be defined equivalently in terms of COO strictly plurisubharmonic functions. Finally, it is useful to note the following special result.

7. THEOREM.

Anyone-dimensional Riemann domain is pseudoconvex.

Proof. Suppose M is a connected one-dimensional Riemann domain with projection P: M -+ e, and write M = U.M. where M. are open subsets of M such that M. is compact and M. s;;; M.+ 1 • For any point a E aM., some open neighborhood Ua of a in M is mapped biholomorphically by P to an open subset P(Ua ) S;;; Co The image P(Ua n M) S;;; e is a domain of holomorphy, as is any open subset of e, and hence is pseudoconvex; so by Theorem M3 the function -log dUanM• is plurisubharmonic in Ua n M •. Since it is clear that -log dM,(z) = -log dUanM,(z) whenever Z E Ua n M. is sufficiently near a, it follows that -log dM , is plurisubharmonic in U where U is the intersection with M. of some open neighborhood of the compact subset aM. of M. The function 1 + IP(z) I is plurisubharmonic in all of M, and since it is strictly positive, there is a constant c > 0 sufficiently large that c(l + IP(z)l) > -log dM,(z) whenever z E M. - U•. The function u. defined by u.(z) = sup( -log dM,(z), c(l

+ IP(z)l»

is then evidently a continuous plurisubharmonic exhaustion function for M., so that M. is pseudoconvex by Theorem 3. Incidentally, once having demonstrated that M. is pseudoconvex, it follows from Theorem 3 that -log dM • is itself a plurisubharmonic exhaustion function for M. An application of Theorem 4 shows then that M is pseudoconvex, to conclude the proof.

o Pseudoconvexity and Dolbeault Cohomology

en in section M that pseudoconvexity is actually equivalent to holomorphic convexity, and hence that the properties of pseudoconvexity established in that section automatically hold for holomorphic convexity as well; the same is true in the case of Riemann domains. The proof involves a rather detailed analysis of some formal properties of the aoperator and an application of some fairly standard techniques from the theory of linear partial differential equations to show that the Dolbeault cohomology groups Jep,q(D) of a pseudoconvex open set D £ en vanish whenever q > O. It then follows from Theorem G14 that D is a domain ofholomorphy as desired. The same argument with minimal modifications shows that these Dolbeault cohomology groups of a pseudoconvex Riemann domain also vanish, leading to the result that a pseudoconvex Riemann domain is a Riemann domain of holomorphy. Rather than repeating the argument or referring back to it in the later discussion of Riemann domains, the proof will be presented in that generality from the beginning; those readers interested only in this result for open subsets of en should not find the minor further generality at all burdensome. Suppose then that M is a Riemann domain with projection P: M -+ en. The image points Z = P(Z) E en provide natural local coordinates at all points Z E M, and differential forms on the complex manifold M will be written solely in terms ofthese coordinates in this section. Thus, a Coo differential form

;Fp,q] the element 0( E [;Fp,q] as just constructed is evidently a well-defined linear mapping S: [l:l;Fp,q] -+ [;Fp,q]. It follows readily from the inequality (20) that S is actually a continuous linear mapping, since IISPilu = lim. 110(. IIu ~ lim V II !:>wu 0(. II w ~ II PII w; it is also in a sense the in verse of l:lwu, at least to the extent that S(!:>wuO() = 0( whenever 0( E ;Fp,q. In terms of this mapping S and the given differential form tP, introduce the continuous linear functional T;: [l:l;Fp,q] -+ C defined by T~(P) = (SP, tP)u, noting that the continuity of T~ follows from the continuity of S. As a consequence of well-known general properties of Hilbert space, there must exist an element () in the Hilbert space [l:l;Fp,q] such that T~(P) = (P, (})w' Of course, since the elements !:>wuO( are dense in [!:>;Fp,q] as 0( ranges over ;Fp,q, the characteristic property of the element () E [!:>;Fp,q] can be restated equivalently as the property that (!:>wuO(, (})w = T~(S!:>wuO() = (0(, tP)u for all 0( E ;Fp,q. The element () E .P$,q-l can be interpreted as a differential form of bidegree (p, q - 1) on M with locally square-integrable coefficients. It is then a simple matter to show that this differential form has the desired properEes. First, whenever 't' E tff/'Mq- 2 and 0( E ;Fp,q, then (a't', !:>wuO()w=(aa't', O()u=O. Thus, 8't'1. l:l;Fp,q, and since () E [l:l;Fp,q], it follows that (a't', (})w = O. Next any element U E tff/'Mq £::: 'p,:,q can be written as a sum U = 0( + 0(' where 0( E [;Fp,q] and 0(' ..1 [;Fp,q], and 0( = lim. 0(. for some differential forms 0(. E ;Fp,q. Since tP E ;Fp,q, it follows of course that(u, tP)u = (0(, tP)u = lim. (0(., tP)u = lim. (l:lwuO(., (})u, in view of the characteristic property of the element (). On the other hand, whenever y E tI/'Mq- 1 , then (!:>wuu, y)w = lim. [(!:>wuO(., y)w + CDwuu - !:>wuO(., Y)w] = lim. (l:lwuO(., y), for lim. (!:>wuu - l:lwuO(., y)w= lim. (u - 0(., ay)u=(O(', ay)u=o since ay E /Fp,q while 0(' 1. [:Fp,q]. Since tI/'Mq- 1 is dense in .P$,q-l, it follows actually that (l:lwuu, Y)w =

.J2

.J2

160

Volume I

Function Theory

limy (l)wucx., y) for all y E Sf!,q-1, in particular that (l)wuO", e)w = lim v (l)wucx v ' e); therefore, (0", (P}u = (l)wuO", e)w, and that concludes the proof, If e E Cl/q satisfies condition (i) of the preceding theorem, then (0", ae)u = (l)wuO", e)w = (0", rP)u whenever 0" E C!'ii; hence, ae = rP, For q > 1, there are a great many differential forms ewith locally square-integrable but nondifferentiable coefficients satisfying condition (i), and these forms are only solutions of the partial differential equation ae = rP in the weak sense specified in the statement of the preceding theorem. But it will be shown that any differential form e having locally square-integrable coefficients and satisfying both conditions (i) and (ii) is essentially a Coo differential form e E Cl/q and hence does satisfy ae = rP in the straightforward sense. The remainder of this section will be devoted principally to the demonstration ofthis regularity theorem. For this purpose it is convenient to use unweighted inner products on the spaces of differential forms on M, inner products (rP, 1/1) defined as in (6) but with weighting factor u = 0, so that (rP, I/I)u = (e-urP, 1/1) = (rP, e-ul/i). It then follows from Lemma 2 that there is a linear differential operator l): cfl/ q+1 -+ cfl/ q such that (arP, 1/1) = (rP, l)I/I) whenever rP E Cl/q, 1/1 E Cl/q+1, and at least one of these two differential forms has compact support. The operator l) has the explicit form (10) with u = v = 0 and hence is a first-order homogeneous linear partial d~fferential operator with constant coefficients. In terms of this operator the differential operator l)wu of Theorem 5 has the form l)wuO" = eWl)(e-uO"). Note that (l)wuO", e)w = (l)(e-uO"), e) and (0", rP)u = (e-uO", rP), so the conclusion of Theorem 5 can be rephrased as the assertion that the differential form satisfies

e

(i)

(l)cx, e) = (cx, rP) whenever cx

(ii)

(ap, e-We) = 0 whenever p E

E

C!'ii (21)

C!'i.r 2

The case q = 1 merits separate treatment, since in that case condition (ii) is vacuous and the desired regularity can readily be deduced from the following result, which is interesting in its own right and is in turn a simple consequence of previously demonstrated results. 6, THEOREM (weak form of the Cauchy-Riemann criterion).

A locally Lebesgue integrable function u in an open subset U ~ en is equal almost everywhere in U to a holomorphic function precisely when Ju u(Z) av(Z)/azj dV(Z) = 0 for I ~ j ~ n and for any Coo function v with compact support in U.

Proof. Whenever oc = and consequently

(u, l)cx) = -

~ J

Lj CXj dZj E Cc'b 1, it follows from (10) that l)cx = - Lj acxj/azj,

r u(Z) aa.a:i~Z) dV(Z)

Ju

Zj

If u is equal almost everywhere in U to a holomorphic function u', then (u, l)oc) = (u', l)oc) = (au', oc) = 0; hence, u evidently satisfies the conditions of the theorem.

o

Pseudoconvexity and Dolbeault Cohomology

161

Conversely, if u satisfies the conditions of the theorem, then (u, nO() = 0 whenever 0( E Iffc~ 1; hence, in particular for any C'" function v with compact support in U and any vector A E en it follows upon setting O(j = Lk aiik iJvjiJzk that

= -

Iv

u(Z)· Lv(Z; A) dV(Z)

If v is a real-valued function, this last equality holds separately for the real and imaginary parts of u, and it then follows from Corollary K18 that the real and imaginary parts of u are equal almost everywhere in U to pluriharmonic functions. Thus, after modifying u on a set of measure zero, it can be supposed that u is a C'X! function in U; but then since 0 = (u, nO() = (au, O() whenever 0( E Iffc~ 1, it follows that au = 0 and hence that u is holomorphic in U as desired. That suffices to conclude the proof. 7. COROLLARY. If M is a pseudoconvex Riemann domain and ifJ there is a differential form 0 E Ifffio such that ao = ifJ.

E

Ifffi 1 satisfies aifJ = 0, then

Proof It follows from Theorem 5 that there exists a differential form 0 of bidegree (p,O) with locally square-integrable coefficients on M such that (nO(, 0) = (0(, ifJ) whenever 0( E Iff!'i}, from (21), and as already noted it is sufficient to show that 0 is equal almost everywhere to a C'X! differential form on M. If A is any point of M and U is an open neighborhood of A such that an open neighborhood of 0 is mapped biholomorphically into en by the projection mapping P: M --+ en and P(U) is a polydisc centered at P(A), then by Dolbeault's lemma, Theorem E3, there is a COO differential form 1/1 of bidegree (p, 0) in U such that ifJl U = al/l. Now (nO(, 0 - 1/1) = (nO(, 0) - (nO(, 1/1) = (0(, ifJ) - (0(, al/l) = 0 whenever 0( E Iff!'i}. Then from (10) it is easy to see that

so it follows from the preceding theorem that each coefficient 01 - 1/11 is equal almost everywhere in U to a holomorphic and hence COO function. Then 01 is also equal almost everywhere in U to a COO function, and the proof is thereby concluded. The general case of the desired regularity theorem cannot so readily be reduced to known results about holomorphic or harmonic functions, since condition (ii) of equation (21) involves the rather arbitrary auxiliary function w, but nonetheless can be demonstrated fairly handily using standard methods from the theory of elliptic partial differential equations. To introduce these methods, consider first an arbitrary linear partial differential operator D: Iff£jq --+ Iff£jq having constant coefficients when expressed in terms of the standard local coordinates on M. For instance, Dt/J may consist in applying the operator jJkjaz~ to one particular coefficient

162

Volume I

Function Theory

of rfo and multiplying all others by zero. It is easy to see that to any such operator D there corresponds an adjoint operator D*: cfl;q -+ cfl;q, another linear partial differential operator with constant coefficients and of the same order as D such that (DIX, fJ) = (IX, D*fJ) whenever IX E cfl;q, fJ E cfl;q, and at least one of these forms has compact support in M. Indeed, a differential form in the standard coordinates can be viewed merely as a collection of CCXl functions, with the inner product (6) in which u = 0, and any differential operator D can be expressed as some combination of the derivatives o/OZj and O/OZk acting on these coefficients. Since the operators OjOZj and -%~ are adjoint as in equation (12), it is a simple matter to calculate the desired operator D*, but the explicit forms will not be needed here, so further details will be omitted. Now let ~p,q be the space of all differential forms of bidegree (p, q) having Lebesgue measurable and locally square-integrable coefficients on M, and more generally for any integer v ~ 0, let 1f".p,q consist of all those forms rfo E ~p,q such that to each linear partial differential operator D: cfl;q -+ cfl;q with constant coefficients and of order ~ v there corresponds another form rfoD E ~p,q for which (rfo, D*IX) = (rfoD, IX) whenever IX E cf!'ii. Thus, "If;,p,q can be viewed as the subspace of ~p,q consisting of those forms for which each coefficient has square-integrable partial derivatives of order ~ v in the sense of distributions, and rfoD = Drfo also in the sense of distributions. The precise meaning of these statements consists of the definitions just given, though. It is a straightforward matter to verify that frfo E "If;,p,q whenever f E cf~o and rfo E "If;,p,q. Next note that to each differential form rfo E "If/rJ',q it is possible to associate the smoothed differential forms rfo. defined for any e > 0 by applying the construction of Definition K9 to each component of rfo when expressed in terms of the natural coordinates on M. Thus, rfo. is a well-defined CCXl differential form of bidegree (p, q) on the open subset M. = {Z EM: "M(Z) > e}, with the notation introduced in Definition H2, and it follows from Lemma K 10 that as e tends to zero, the differential forms rfo. converge to rfo in the norms (22) for any compact subsets K c M. If rfo E ~p,q, I/J E forms has compact support contained in M, then

=

t f- r

I,J

=

ZeM

Jwed

~p,q,

and at least one of these

rfol,J(Z + eW)O'(W)ifll,AZ) dV(W) dV(Z)

t f- JW'ed r_ rfol,AZ')O'(W')ifll,AZ' + eW') dV(W') dV(Z')

I,J

Z'eM

with the obvious change of coordinates

Z'= Z + eW, W' = -

W. If rfo E Gf;q, it is

o

Pseudoconvexity and Dolbeault Cohomology

163

quite obvious from the formula of Definition K9 that D(r/J.) = (Dr/J). for any linear partial differential operator D with constant coefficients, so fOT a COO differential form r/J, the notation Dr/J. is unambiguous. On the other hand, if r/J E "If"vp,q and if D is a linear partial differential of order ~ v with constant coefficients, then (D(r/J.), IX) = (r/J.. D*IX) = (r/J, D*IX.) = (r/JD, IX.) = «r/JD).. IX) whenever IX E 8!'ii has support contained in M •. Consequently, D(r/J.) = (r/JD). at least in the open subset M. c M. These constructions and observations can be applied to the problem at hand by using the following elementary version of Sobolev's lemma.

8. LEMMA. If r/J E n~o "If"vp,q, then r/J is equal almost everywhere on M to a COO differential form in 8l;q. Proof. For any point A EM choose an open neighborhood .1 of A in M such that .1 is mapped biholomorphically by the projection P to an open polydisc P(.1) centered at A = P(A) in en. When considering just the neighborhood .1, it can be identified with the polydisc P(.1) without leading to any confusion. Also choose a COO function p on M such that p is identically equal to one in an open neighborhood of A and the support of p is a compact subset of .1. Then pr/J E "If':,p,q and pr/J has compact support in .1. The smoothed differential forms 1/1. = (pr/J). are COO differential forms with compact supports in .1 whenever 6 is sufficiently small, and 111/1. - Pr/JII;1 tends to zero as 6 tends to zero where the norm is that defined as in (22). It will be demonstrated that for any linear partial differential operator D: 8l;q -+ 8l;q with constant coefficients, the differential forms DI/I. converge uniformly on .1 as 6 tends to zero. That evidently implies that the differential forms 1/1. converge uniformly on .1 to a COO differential form 1/1 E 8f.,q as 6 tends to zero, and since 111/1 - Pr/JII;1 = lim._ o 111/1. - Pr/JII;1 = 0, then pr/J is equal to 1/1 almost everywhere in .1. Thus, r/J is equal to a COO differential form almost everywhere in a neighborhood of A, and that gives the desired result. To conclude the proof in this manner, consider in addition to D the linear mapping Do: 8l;q -+ 8l;q defined by applying the differential operator iP"/OX1 ... OX"OY1 ... oY" to each coefficient of a form in 8l;q, where as usual Zj = Xj + iYj. Since DoD(I/Io) = (pr/J)1joD in .1 whenever 6 is sufficiently small, it is also the case that IIDoD(I/I.) - (pr/J)DDo 11;1 converges to zero as 6 tends to zero and hence that the differential forms DoD(I/I.) form a Cauchy sequence in this norm. However, for any points B E P(.1) and C ¢ P(.1) in e" and for any COO function f of compact support in P(.1), note that

nv

~

I

IDof(Z)1 dV(Z)

from which it follows that

~ 1.111/2 II DoJII;1

164

Volume I

Function Theory

D"'. D"'.

The coefficients of the differential forms are therefore also a Cauchy sequence in the supremum norm over ~; hence, converges uniformly on ~ as e tends to zero. As already noted, that suffices to conclude the proof. 9. THEOREM. If M is a pseudoconvex Riemann domain and t/J E 8fjq for q> 0 satisfies at/J = 0, then there is a differential form e E tfffjq-l such that ae = t/J. Proof. Since the special case of the theorem in which q = 1 was already established in Corollary 7, it can be assumed for the remainder of this proof that q > l.1t follows from Theorem 5 that there exists a differential form E 1fQp,q-l satisfying conditions (21). The proof will proceed by showing by induction on v that this differential form e actually belongs to 1f:,P,q-l for all indices v ~ O. It then follows from Lemma 8 that is equal almost everywhere on M to a COO form E 8fjq-l, and as already noted this form e' has the desired property that ae' = t/J. Suppose therefore that eE 1f:,P,Q-l for some index v ~ O. Condition (i) of equation (21) asserts that rDa, e) = (a, t/J) whenever a E tff!ii, while condition (ii) asserts that (api, e-We) = 0 whenever pi E tff!Mq- 2 for some real-valued Coo function w on M. To rewrite condition (ii), set p = e-Wp', and note then that 0 = (e-Wa(eWp), e) = (ap, e) + (aw 1\ p, e). But since

e

e

e'

(ow

1\

p, e) = =

where",

E 1f:,p,q-2

-

L* L

I,I,K k

i

M

(-l)P ow .:l- PI,I sign(kKJ)()l,J dV(Z) uZk

(P, -"') is the contracted differential form

'" = owJe = (_l)P+l

ow L* Lk ~ eI,kl dZI 1\ dZI uZk

(23)

1,1

it follows that (ap, e) = (P, "') whenever p E 8!Mq- 2 • Now let D: 8fjq-l -+ 8fjq-l be an arbitrary linear partial differential operator of order ~ v with constant coefficients on M. For any fixed a E tff!Mq and all sufficiently small e, it follows from condition (i) that (a, aDe.) = (D*!)a, e.) = (tW*a., e) = (D*a., t/J) = (a, Dt/J.); hence, aDe. = Dt/J. on any compact subset of M for all sufficiently small e. Similarly, for any fixed p E tS'!MQ- 2 and all sufficiently small e, it follows from condition (ii) and the observation", E 1f:,P,Q-2 that (p, !)De.) = (D*ap, e.) = (aD*P., e) = (D*P.. "') = (P., ",D) = (P, (",D).); hence, !)DO. = (",D). on any compact subset of M for all sufficiently small e. It results therefore upon recalling Lemma KlO that as e tends to zero the differential forms aDO. converge to Dt/J and the differential forms !)DO. converge to ",D in the norms (22). Next for any compact subset K £; M, choose a Coo function p such that p is identically equal to one on K and the support of p is a compact subset of M. It is clear that a(pDe.) = p' aDO. + ap 1\ DO., and a simple calculation using (lO) shows that !)(pDe.) = p' !)DO. + opJDO. with the notation as in (23). Consequently, from the induction hypothesis and the preceding observations, the differential forms a(pDfJ.) and l)(pDO.) also converge in the norms (22) as

o

Pseudoconvexity and Dolbeault Cohomology

165

6 tends to zero. Since these latter differential forms have compact support, they even converge in the norm I . II corresponding to K = M in (22), and from Lemma 3 for the special case u = v = w = 0 it follows that functions arising by applying O/OZk to any coefficient of the differential forms pD(). also converge in this norm as 6 tends to zero. Upon taking complex conjugates, precisely the same arguments applied to the operators 0 and 15 also show that the functions arising by applying O/OZk to any coefficient ofthe differential forms pD. also converge in this norm as 6 tends to zero, or alternatively the same result follows upon noting as an easy double application of (12) that lIof/ozkll = I of/ozk I for any CCX) function f with compact support. An evident consequence of this is that for any linear partial differential operator D': tCli q - 1 --+ tCli q - 1 of order ~ v + 1 with constant coefficients, the differential forms D'(). converge in the norm (22) for the given compact set K as 6 tends to zero, since p(Z) = 1 whenever Z E K. The set K was arbitrary, and since lim..... o D'(). = ()D E "If"oP,q-l, it follows that () E "II'/:·t; that demonstrates the desired induction step and thereby concludes the proof.

The result of the preceding theorem can of course be restated as follows.

The Dolbeault cohomology groups of a pseudoconvex open subset M £; or of a pseudoconvex Riemann domain M satisfy .Jft'p,q(M) = 0 whenever q > O.

10, COROLLARY.

The argument in this section follows that in Hormander [32].

en

p Pseudoconvexity and Holomorphic Convexity

The results of the preceding section can now be applied to show that pseudo convexity and holomorphic convexity are indeed equivalent conditions. To begin with the slightly easier case of open subsets of en, the relevant results can be summarized as follows.

1. THEOREM. (i) (ii) (iii) (iv)

For an open subset D £

en the following conditions are equivalent:

D is a domain of holomorphy. D is holomorphically convex. D is pseudoconvex. yt'o,q(D) = 0 for 0 < q < n.

Proof. The equivalence of conditions (i) and (ii) was demonstrated in Theorem GS. It is obvious that (ii) implies (iii), as noted at the beginning of section M; it follows from Corollary 010 that (iii) implies (iv); and it follows from Theorem G14 that (iv) implies (i), thereby completing the circle of implications and concluding the proof.

Since it has been demonstrated that pseudoconvexity and hoi om orphic convexity are equivalent conditions for an open subset D £ en, and that both are in turn equivalent to the condition that D be a domain ofholomorphy, there is a wide variety of criteria available for demonstrating that an open subset D £ e" is a domain of holomorphy, including both the criteria discussed in section G and the criteria for pseudoconvexity discussed in Section M. In addition, various properties of pseudoconvexity discussed in section M extend immediately to holomorphic convexity. Although the extensions are trivial, the final results are sufficiently important to warrant repeating here explicitly.

2. COROLLARY.

If D is an open subset ofe" and if each point A E aD has an open neighborhood UA such that UA (') D is a domain of holomorphy, then D is also a domain of holomorphy. Proof.

M8.

This is an immediate consequence of the preceding theorem and Theorem

P Pseudoconvexity and Holomorphic Convexity

167

This shows that the property that an open subset of en be a domain of holomorphy is purely a local property of the boundary of that set, a very important result in the theory of holomorphic functions of several variables. As already mentioned, the problem of demonstrating the local character of the condition that an open subset of en be a domain of holomorphy is usually called the Levi problem. 3. COROLLARY (theorem of Behnke and Stein). If Dv are domains of holomorphy in en with Dv ~ Dv+1' then their union D = Dv is also a domain of holomorphy.

Uv

Proof. M6.

This is an immediate consequence of the preceding theorem and Theorem

Theorem 1 is important not only for showing the equivalence of hoi om orphic convexity and pseudo convexity for subsets D ~ en, with such consequences as illustrated in the two preceding corollaries, but also for showing the equivalence of holomorphic convexity and the condition that Je°,q(D) = 0 for 0 < q < n, with such consequences as will be illustrated in the next two corollaries. The vanishing of these Dolbeault cohomology groups for open polydiscs was used quite essentially in the discussion in sections E and F, and it is of comparable utility in studying function theory in general domains of holomorphy. A simple illustrative example is the following. 4. COROLLARY. If D ~ en is a domain of holomorphy, h: D -. e is a nonsingular holomorphic mappingfromD intoe, and M ~ Dis thecomplexsubmanifold M = {Z E D: h(Z) = O}, then for any function f holomorphic on M there exists a function 9 holomorphic in D for which glM = f Proof.

This is an immediate consequence of the preceding theorem and Theorem

E7. It is possibly worth noting that the preceding corollary is not true for arbitrary subsets D ~ en. For instance, if D is the complement of the closed unit ball in e 2 and M = {Z ED: Z2 = O}, then M is really just the complement of the closed unit disc in the plane of the complex variable Z l ' There are on M holomorphic functions that cannot be extended to holomorphic functions in the entire plane of the variable Z 1, and such a function cannot be extended to a holomorphic function in D, since by Hartogs's extension theorem, Theorem E6, any function holomorphic in D necessarily extends to a function holomorphic in all of e 2 • In the situation of Corollary 4 the hypothesis that the submanifold M can be defined by a single function holomorphic throughout all of e was very definitely used in the proof of Theorem E7 but is not really essential. Although a considerably more general result will be demonstrated later, it is perhaps worth noting here as another consequence of Theorem 1 how this hypothesis can be eliminated at least to some extent. Indeed, a complex submanifold M of dimension n - 1 in a domain of holomorphy D £; en can always be written as the zero locus of a nonsingular holomorphic mapping h: D -. en provided there is no topological obstruction to doing so. To make clear what is meant by a topological obstruction in this context,

168

Volume I

Function Theory

note that there obviously exist a covering of D by open subsets ~ and nonsingular holomorphic mappings hj: ~ --. e such that M n ~ = {Z E ~: hj(Z) = O}. The quotients hij = hdhj are nowhere vanishing holomorphic functions in the intersections Ui n Uj and satisfy hiiZ)hji(Z) = 1 whenever Z E Ui n ~ and hij(Z)hjk(Z)hki(Z) = 1 whenever Z E Ui n ~ n Uk' so they form a multiplicative analogue of the Cousin data of Theorem E7. If there does exist a nonsingular holomorphic mapping h: D --. e such that M = {Z ED: h(Z) = O}, then the quotients gi = hdh are holomorphic nowhere vanishing functions in the sets Ui and hij = gdgj in Ui n ~. To say that there is no topological obstruction just means that there exist at least Coo nowhere vanishing functions gi in the sets Ui such that hij = gdgj in Ui n ~. There do not always exist such functions gi' so the properties of the multiplicative Cousin data are not totally parallel to the properties of the additive Cousin data considered earlier. It is constructive to try to reduce the multiplicative Cousin data to additive Cousin data by taking logarithms, and thereby to reinterpret the topological obstruction involved; but that will be left for the interested reader to pursue, since the same analysis in more general terms will appear in Volume III after the introduction of sheaf theory.

5. COROLLARY. If M is a complex submanifold of dimension n - 1 in a domain of holomorphy D s en, then there exists a nonsingular holomorphic mapping h: D --. e such that M = {Z ED: h(Z) = O} provided that there is no topological obstruction. Proof. As noted, there exist a covering of M by open sets ~, which will here be assumed simply connected, and nonsingular holomorphic mappings hj: ~ --. e such that M n ~ = {Z E ~: hj(Z) = O}. The assumption that there is no topological obstruction implies that there also exist Coo nowhere vanishing functions gj in the sets ~ such that gdgj = hdhj in Ui n ~. Since the sets ~ are simply connected, it is possible to choose single-valued branches of the functions log gj in each ~, and evidently log gi - log gj is holomorphic in Ui n ~. It is therefore possible to introduce a well-defined a-closed Coo differential form f/J on D by setting f/J(Z) = a log gi(Z) whenever Z E Ui' It follows from Theorem 1 that Jlt'0,1 (D) = 0, so there is a Coo function 9 in D such that ag = f/J. Now the functions /; = e-gg i are then hoi om orphic and nowhere vanishing in Ui' and /;/jj = gdgj = hdhj in Ui n ~. The function h defined by setting h(Z) = hi(Z)//;(Z) whenever Z E Ui is then a holomorphic function in D defining the submanifold M as desired, thus concluding the proof.

It is worth observing explicitly here that the proofs of the preceding two corollaries really rest solely on the condition that Jlt'0,1 (D) = O. Therefore, both corollaries actually hold for any complex manifold D for which Jlt'0,1 (D) = o. Examples. It is easy to see that the preceding corollary is not true for arbitrary open subsets of en when n> 1. For instance, let D be the union of the open ball DI = {ZEe 2 : IIZII 1 satisfying (iv), and suppose that the desired result has been demonstrated for Riemann domains of dimensions < n. It will first be shown that for any compact subset K £; M and any poly radius R, necessarily bM,R(K) = bM,R(KM). If that is not the case, there exist a compact subset K £; M and polyradius R such that bM,R(K) > bM,R(K M); hence, there is a point A E KM for which bM,R(A) < bM,R(K) = b. Now for any function f E (!)M the argument of Lemma G4 shows that the power series expansion about the point P(A) E en of the function fA = f 0 (PIL\M(A; bM,R(A)R»-l actually converges in the polydisc L\(P(A); bR) and therefore that PM,R(f; A) ;;:;; b > bM,R(A), where P: M --+ en is the projection mapping and the radius of convergence PM,R(f; A) is as in Definition 11. By Lemma 14 that means that the Riemann domain M admits a properly larger holomorphic extension E, with a projection mapping that will also be denoted by P. Choose a point BEaM nE and an (n - I)-dimensional linear subspace L £; en tlirough P(B). The submanifold E' = P-i(L) £; E is clearly a Riemann domain with the projection P' = PIE': E' --+ L ~ en- i , as is the submanifold M' = P-i(L) £; M, and it can be assumed that L is so chosen that BEaM' n E' as well. Just as in the proof of Theorem G14 it can be shown that -*,o,n(M') = 0 for 0 < q < n - 1, so by the inductive hypothesis M' is holomorphically convex and hence is a domain of holomorphy. There must consequently exist a holomorphic function f E (!)M' that cannot be extended as a holomorphic function to E'. However, since -*,o,l(M) = 0, it follows from Theorem E7 that f extends to a holomorphic function on M, and hence necessarily extends further to a holomorphic function on E :2 E'. That is a contradiction, from which it follows as desired that bM,R(K) > bM,R(KM) for any compact subset K £; M and polyradius R. From this result it is not at all difficult to show that M is holomorphically convex. Indeed, if M is not holomorphically

174

Volume I

Function Theory

convex, it follows from Lemma 19 that there exist a compact subset K £; M and a point Zo E C n such that p-l(ZO) n KM contains infinitely many distinct points A v. Now if Ll £; C n is an (n - I)-dimensional linear subspace passing through Zo and Ml = P-l(Ld, then as already noted Ml is a Riemann domain for which ;¥,O,q(Md = 0 whenever 0 < q < n - 1, and any holomorphic function on Ml extends to a holomorphic function on M. Then if L2 £; Ll is an (n - 2)-dimensional linear subspace passing through Zo and M2 = P-l(L 2 ), the same argument shows that M2 is a Riemann domain for which ;¥,O,Q(M2 ) = 0 whenever 0 < q < n - 2, and any holomorphic function on M2 extends to a holomorphic function on Ml and thence to a holomorphic function on M. Continuing in this way shows eventually that any function on the discrete subset P-l(ZO) extends to a holomorphic function on M. In particular, therefore, there exists a holomorphic function f E (!)M such that limv If(Av)1 = 00 where {Av} = P-l(Zo)nK M; but that is impossible, since If(Av)1 ~ IlfilK by the definition of the holomorphically convex hull K M. That contradiction shows that M is holomorphically convex as desired. Note incidentally that in proving that (iv) implies (i) it was actually demonstrated that for any point Zo E P(M) there exists a holomorphic function on M taking any preassigned values at the points P-l(ZO)' Therefore, holomorphic functions separate points on any Riemann domain satisfying (iv), and with that the proof is concluded. Just as for open subsets ofC n so also for Riemann domains do the criteria for and properties of pseudoconvexity carryover to provide criteria for and properties of hoI om orphic convexity. A Riemann domain is a Riemann domain ofholomorphy precisely when it is holomorphically convex, without the assumption that holomorphic functions separate points. Moreover, since the Dolbeault cohomology groups in positive dimensions vanish on Riemann domains of holomorphy, Corollaries 4 and 5 also hold for Riemann domains of holomorphy. Finally, the other criteria of Theorem 113 also hold without the necessity of assuming that holomorphic functions separate points, as follows.

8. THEOREM.

For a Riemann domain M the following conditions are equivalent:

(i) M is a Riemann domain of holomorphy. (ii) For any discrete sequence of points Av E M there exists a function f E (!)M for which lim supv If(Av)1 = 00. (iii) M admits no properly larger holomorphic extension. (iv) For any polyradius R and any compact subset K £; M, necessarily bM,R(K) = bM R(K M )· (v) There exists a fixed polyradius R such that bM,R(K) = bM,R(KM)for any compact subset K £; M. (vi) dM(K) = dM(K M) for any compact subset K £; M.

Proof. That (i) implies (iv) follows from Theorem 17. It is, of course, obvious that (iv) implies (v), and it is a purely formal consequence of the definition of these distance functions as in Corollary G6 that (iv) implies (vi). Essentially the same proof shows that either (v) or (vi) implies (i); so for this purpose let d denote either bM,R

P Pseudoconvexity and Holomorphic Convexity

175

for some fixed polyradius R or dM, and suppose that d(K) = d(KM) for any compact subset K ~ M. It will then be demonstrated that M is pseudoconvex in the sense of Hartogs, which by Theorems N3 and N7 implies that M is a Riemann domain of holomorphy. If ~: [0, 1] x A(O; 1) -+ en is a continuous mapping from the compact subset [0, 1] x .:\(0; 1) ~ IR x e into en such that F is holomorphic in .:\(0; 1) for each fixed point of [0, 1], and if G: ([0, 1) x A(O; 1» u (1 x 0.:\(0; 1» -+ M

(1)

is a continuous mapping such that P 0 G = F where P: M -+ en is the projection mapping, then clearly K = G([O, 1] x 0.:\(0; 1» is a compact subset of M and KM contains the entire image of the mapping (1). But since d(KM) = d(K) > 0 by hypothesis the mapping G obviously extends to a continuous mapping G: [0, 1] x A(O; ) -+ M. That means that M is pseudoconvex in the sense of Hartogs, and consequently it has been demonstrated that conditions (i), (iv), (v), and (vi) are equivalent. Next, to show that (i) implies (ii), note that by Theorem 7 if a Riemann domain M satisfies (i), then Mis holomorphically convex; but the proof of Theorem G7 then shows that M also satisfies (ii). It is obvious that (ii) implies (iii). Finally to demonstrate that (iii) implies (i), suppose to the contrary that M is a Riemann domain satisfying (iii), but that M is not a Riemann domain of holomorphy. Since it has already been shown that (i) is equivalent to (iv), there must be some compact set K ~ M and polyradius R for which bM R(K) > bM R(K M). There is thus a point A E KM for which bM,R(A) < bM,R(K) =~. Then fr~m Lemma G4 it follows that PM,R(f; A) ~ b for every f E (I)M, and that in turn by Lemma 4 shows that M admits a properly larger holomorphic extension, contradicting the hypothesis that M satisfies (iii). With that contradiction it has been shown that conditions (i), (ii), and (iii) are also equivalent, thereby concluding the proof. As a final comment it is perhaps worth noting that if M is a Riemann domain for which holomorphic functions separate points, then M is a Riemann domain of holomorphy precisely when M = E(M). This is really just a special case of the equivalence of conditions (i) and (iii) in the preceding theorem, the hypothesis that holomorphic functions separate points being required here in order that the envelope of holomorphy E(M) be well defined.

a Plurisubharmonic and Holomorphic Functions

Plurisubharmonic functions played an important role in the preceding discussion of domains ofholomorphy, and they have many more relations to and consequences for hoi om orphic functions, just two of which will be discussed briefly here. One of these is a useful extension of Riemann's removable singularities theorem. It was noted at the beginning of section M that If I is plurisubharmonic whenever f is holomorphic. Actually it follows immediately from Jensen's inequality in several variables, Theorem A8, that loglfl is plurisubharmonic whenever fis hoi om orphic. This is a somewhat stronger result, since the real exponential function is convex and monotonically increasing, so by Theorem KS(d) the plurisubharmonicity ofloglfl implies that of If I = exp loglfl. The zero locus of f is the set of points at which the function loglfl takes the value -00. The major role played by the zero sets of holomorphic functions thus suggests introducing the following notion. 1. DEFINITION. A subset X of an open set D ~ en is called a pluripolar set in D if for each point A ED there are an open polydisc A(A; R A ) and a plurisubharmonic function UA in A(A; R A ) such that U A is not identically equal to -00 and

..."

Note that a pi uri polar set X in D is necessarily a proper subset in each connected component of D; indeed, it follows immediately from Theorem K6 that X is a subset of D of Lebesgue measure zero. The pluripolar sets can be viewed as playing for plurisubharmonic functions roughly the same role that thin sets play for hol0 7 morphic functions. Actually since loglfl is plurisubharmonic whenever f is holomorphic, it is evident that any thin set in D is also a pluripolar set in D. The converse assertion is false, for the class of pi uri polar sets is considerably more extensive than the class of thin sets. For example, suppose that Xl' X 2, X 3, ... are pluripolar sets in a connected open subset D ~ en, and that Xy ~ {Z ED: uy(Z) = - oo} where U y are plurisubharmonic functions that are not identically equal to -00 in D; then X = Uy Xy is a pluripolar set in D. Indeed choose a sequence of open subsets Dy ~ D such that Dy is a compact subset of D, that Dy ~ Dy+ 1 , and that Uy Dy = D; and choose a point A E D such that uy(A) # - 00 for all indices v, noting that there

Q Plurisubharmonic and Holomorphic Functions

177

certainly exists such a point since u. is equal to - 00 only on a set of Lebesgue measure zero. By adding a suitable real constant to u. it can be assumed that u.(Z) ~ 0 whenever Z = A or ZED., since A u D. is compact and hence the mapping u. is bounded from above on that set. By multiplying u. by a suitable positive real constant, it can also be assumed that 0 ~ u.(A) ~ - 2-'. Now the mappings v. = U 1 + ... + u. are plurisubharmonic functions on D, and whenever v ~ J.l this sequence of functions is monotonically decreasing on the subset DIl , since u. ~ 0 on DIl ~ D. by construction. The limit function u = lim. v. = L. u. is therefore a plurisubharmonic function in D, and this function is not identically equal to - 00 since 0 ~ u(A) = L. u.(A) ~ L. - 2-' = - 1. Since moreover it is evident that X. ~ {Z ED: u(Z) = - oo} for each index v, the same is true for the union X = U. X. and hence X is a pluripolar subset of D as desired. In particular, if f. are holomorphic functions that are not identically equal to zero in a connected open set D ~ en and if X. ~ {Z ED: f.(Z) = OJ, then the union X = U.X. is a pluripolar set in D but is not necessarily a thin set in D. Thus, any countable subset of an open set D ~ e is a pluripolar set, but can only be a thin set if it is discrete. Even a countable dense set is a pi uri polar set, so clearly the closure of a pluripolar set is not necessarily a pluripolar set. It should be mentioned here in passing that the assumption that the pluripolar sets X. above are defined by global plurisubharmonic functions u. in D is not really essential, since B. 10sefson has shown that any pluripolar set in an open subset D ~ en can be defined in terms of a global plurisubharmonic function in D. The proof of this latter assertion will not be given here. Thin sets have thus far been considered only in connection with holomorphic extension theorems, in particular in Theorem D2. There are analogous results for plurisubharmonic functions and pluripolar sets, but since such results involve extensions of plurisubharmonic functions from the complement of a pluripolar set X in an open subset D ~ en to all of D, and since plurisubharmonic functions have been defined only in open subsets of en, it is apparent that only closed pluripolar sets X playa role in this discussion. A useful preliminary result in this direction is the following. If u: D -+ [ -00,00) is an upper semicontinuous mapping in an open subset D ~ e", if x is a closed subset of D, and if u is a plurisubharmonic function in D - X and u(Z) = - 00 whenever Z E X, then u is necessarily a plurisubharmonic function throughout D.

2. LEMMA.

Proof.

For any nonnegative integer v, introduce the function u. in D defined by u.(Z)

= sup(u(Z), - v)

This function is of course plurisubharmonic in the open set D - X by Theorem KS(d). On the other hand, since u is upper semicontinuous, D. = {Z ED: u(Z) < - v} is also an open subset of D, and since u.(Z) = - v whenever ZED., it follows that u. is also plurisubharmonic in D•. Since X ~ D. by hypothesis the open subsets D - X and D. cover D, and since plurisubharmonicity is a local property it follows that u. is plurisubharmonic throughout D. Now the functions u. are a montonically

178

Volume I

Function Theory

decreasing sequence converging to u at each point of D, so it follows from Theorem K5(e) that u is plurisubharmonic in D as desired; that suffices to conclude the proof. 3. THEOREM. If X is a closed pluripolar set in an open subset D ~ en, then a plurisubharmonic function u in D - X extends to a plurisubharmonic function in all of D if and only if lim SUPz-+A.zeD-XU(Z) < 00 at each point A E X, and the extension is a uniquely determined plurisubharmonic function in D.

Proof. If there is a plurisubharmonic function v in D such that v(Z) = u(Z) whenever ZED - X, then lim sUPZ-+A.ZeD-X u(Z) = v(A) < 00 at each point A E X. Moreover, since X is a subset of Lebesgue measure zero, it follows from Theorem K15 that v(A)

= lim

I~(A; 8)1-

= lim

I~(A; 8)1- 1

£-+0

£-+0

1

Jr4(A;£) v(Z) dV(Z)

r

u(Z) dV(Z)

J(D-X)f""I4(A;£)

at each point A E D; hence, the plurisubharmonic function v is uniquely determined by u. On the other hand, suppose that u is plurisubharmonic in D - X and that lim SUPZ-+A.ZeD-XU(Z) < 00 at each point A E X. Since X is a pluripolar set, for each point A E X there are an open polydisc ~A = ~(A; RA ) and a plurisubharmonic function WA in ~A such that WA is not identically equal to - 00 and X n ~A ~ {Z E ~A: wA(Z) = -oo}. Then for any 8> 0, introduce the mapping v~: ~A --. [-00, +(0) defined by if Z E (D - X) n if Z E X n~A

~A

This mapping is obviously upper semicontinuous in the open subset (D - X) n ~A' while if B E X n ~A it follows from the hypothesis on u and the definitions of WA and VA that lim SUPZ-+BV~(Z) = lim SUPZ-+B.ZeD-X(U(Z) + 8WA(Z» = -00 = v~(B); consequently, v~ is upper semicontinuous throughout ~A- Since v~ is plurisubharmonic in (D - X) n ~A and v~(Z) = -00 whenever Z E X n ~A' it follows from Lemma 2 that v~ is actually plurisubharmonic in all of ~A' The functions 8WA and v~ are therefore both locally Lebesgue integrable in ~A' since they are not identically equal to - 00. The function u must consequently also be locally Lebesgue integrable in ~A' irrespective of its definition on the subset X n ~A of measure zero. Moreover, it is evident that lim£-+o v~ = u in the topology of local L 1 -convergence; hence, by Corollary K17 the mapping u is equal to a plurisubharmonic function almost everywhere in ~A' This plurisubharmonic function coincides with u in (D - X) n ~A and is uniquely determined as demonstrated in the first part of the proof. So since this holds for all points A E X, it follows that u extends to a plurisubharmonic function in D as desired. That suffices to conclude the proof.

Q

Plurisubharmonic and Holomorphic Functions

179

en,

then

4. COROLLARY. If X is a closed pluripolar set in a connected open subset D D - X is also connected.

~

Proof. If D - X = U 1 U U2 where U 1 , U2 are disjoint nonempty open sets, then the function u in D - X defined by u(z)

-oo ={ 0

is plurisubharmonic in D - X, and lim sUPZ-+A.ZeD-X u(Z) < 00 at each point A E X. It then follows from the preceding theorem that u extends to a plurisubharmonic function in D; but that is impossible, since the extension would be a plurisubharmonic function not identically equal to - 00 in a connected open subset D ~ en but equal to - 00 in a subset U1 ~ D of positive measure, contradicting Theorem K6. That contradiction shows that D - X must be connected as desired. If X is a closed pluripolar set in an open subset D ~ en and f is a uniformly bounded holomorphic function in D - X, then there is a unique holomorphic function jin D such thatj(Z) = f(Z) whenever ZED - X.

5. COROLLARY.

Proof. The functions ± Re fare pluriharmonic and hence plurisubharmonic in D - X, and since both are bounded from above in D - X by assumption it follows from the preceding theorem that both extend to plurisubharmonic functions in D. The extensions are the negatives of one another in D - X, and since X is a subset of D of Lebesgue measure zero, it follows from Theorem KI5(i) that the extensions are the negatives of one another throughout D. Thus, if u is the extension of Re f, then both u and - u are plurisubharmonic in D and hence u is pluriharmonic in D. Similarly, 1m f extends to a pluriharmonic function v in D. Now j = u + iv is a COO function in D and aj = 0 in the dense open subset D - X ~ D, so by continuity aj = 0 in D and hencejis hoI om orphic in D. This provides an extension of the function f as desired, and since D - X is a dense open subset of D, the uniqueness ofj follows immediately from the identity theorem, Theorem A3. That suffices to conclude the proof.

As in the discussion of the corresponding result for thin sets, so also here the theorem can be restated as the assertion that any function that is holomorphic outside a closed pluripolar subset of an open set D ~ en and locally bounded in D has a unique extension to a holomorphicfunction in D. Since pluripolar sets are more general than thin sets, this result is an extension of Theorem D2. An immediate application is the following quite useful and perhaps surprising result. 6. COROLLARY (Rado's theorem). If f is a continuous complex-valued function in an open set D ~ en and if f is holomorphic in the open subset D - X ~ D where X = {Z ED: f(Z) = O}, then f is necessarily holomorphic throughout D. Proof. It follows immediately from the hypotheses that the mapping log If I: D -+ [ - 00, + 00) is upper semicontinuous and that log If I is plurisubharmonic in D - X

180

Volume I

Function Theory

and log If(Z} I = - 00 whenever Z E X. But then by Lemma 2 the function log If I must be plurisubharmonic throughout D. The closed subset X = {Z ED: 10glf(z)l = - oo} is therefore a pluripolar set in D, and it then follows from the preceding corollary that the restrictionflD - X extends to a holomorphic function in D. This extension must coincide with f by continuity, so that f is holomorphic in D as desired. That suffices to conclude the proof. The discussions of plurisubharmonic functions have from time to time involved those of the special form u(Z} = 10glf(Z}1 for a holomorphic functionf; they are a very interesting special class of plurisubharmonic functions, which are in a natural sense extreme points in the convex cone of all the plurisubharmonic functions. This observation is essentially an interpretation of the following simple result. 7. LEMMA. If U1 and U2 are plurisubharmonic functions in an open polydisc .1\(0; R} about the origin in ([:" and if u 1 (Z) + U2(Z} = loglzll, then ui(Z} = Re/;(Z} + ciloglzti where /;(Z) are holomorphic in .1\(0; R} and Ci are positive real constants. Proof. Write .1\(0; R} = .1\(0; r 1 } x .1\(0; R"} C (;1 X (;n-l as usual, and introduce the submanifold V = {Z E .1\(0; R}: ZI = O} c .1\(0; R}. Note that -u 1 (Z} = U2(Z) + Re log liz 1, so that - U 1 is also plurisubharmonic in .1\(0; R} - V; hence, by Theorem K5(h} the function U 1 must actually be pluriharmonic in .1\(0; R} - V, and for the same reasons so, of course, is u 2 • The differential form OUj for j = 1 or 2 is then a closed holomorphic I-form in .1\(0; R} - V, just as in the proof of Theorem K3. If 2niaj is the value of the integral of this differential form oUj along any closed path in .1\(0; R} - V encircling V once, then the function

is evidently a single-valued holomorphic function in .1\(0; R} - V, for any fixed base point A E .1\(0; R} - V. Now d(uj - hj - aj log ZI -hj -li.i log zd = 0 as in the proof of Theorem K3, so that

for some real constant bj. Here Re(aj log ZI} = (Re aj) 10glz11 - (1m aj) arg ZI must be single-valued in .1\(0; R} - V, since uj(Z} and hiZ} are; but that can only be the case when 1m aj = 0, so that aj is also real and (I) For any fixed point Z" E .1\(0; R"} C ([:"-1 the function hj (zl' Z") is holomorphic in the variable ZI in the disc .1\(0; rd except possibly at the origin. However, uj(Zl> Z"} is upper semicontinuous and so is bounded above near the origin in Z 1. Hence, from (I) it is evident that

Q Plurisubharmonic and Holomorphic Functions

181

for 0 < IZ11 < 0 and nowhere-vanishing holomorphic function f1 in U. Now

for any point Z E U, where ( -l/v) loglf1 (Z)I is pluriharmonic in U, so it follows immediately from the preceding lemma that (1/v)u 1 (Z) - (l/v) loglf1(Z)1 - C 1 loglz11 and (1/v)u 2 (Z) - c2 10glz 1 1are pluriharmonic in U for some uniquely determined positive constants cj . Consequently, uj(Z) - cj 10glf(Z)1 are pluriharmonic in U for these constants. This is the case for the same constants cj near any other point Z of V, since Vis connected; thus, uj - cj loglfl is pluriharmonic near Vas well as in the complement of V, and that suffices to conclude the proof. The hypothesis of the preceding theorem is unnecessarily restrictive, simply because the discussion of the zero loci of hoiomorphic functions has not been

182

Volume I

Function Theory

carried out far enough yet. The proper hypothesis, in the terminology to be established in Volume II, is that the zero locus of f is an irreducible holomorphic subvariety of the envelope ofholomorphy of D. That means that a dense open subset of the zero locus off is a connected submanifold of dimension n - 1 in the envelope ofholomorphy, so that the argument of the preceding theorem can be applied, while the remainder of the zero locus is a subvariety of smaller dimension and hence a removable singularity set for plurisubharmonic functions. The details will be left as an exercise to be completed when the relevant material from Volume II has been read, since the result will not be used here. The connectivity of the zero locus was used in the proof of the theorem, and irreducibility is essential. If the zero locus is reducible in the envelope of holomorphy, then f = fd2 for some holomorphic functions./j with distinct zero loci, so that loglfl = 10glfl1 + loglf21 and loglfl is not extremal. . There remains the question whether the converse of the preceding theorem holds-that is, whether the extremal plurisubharmonic functions are all of the form loglfl. The interest in extremal functions is, of course, motivated by the KreinMilman theorem, which asserts that under suitable conditions all points in a convex cone can be expressed as convex linear combinations of extreme points of the cone. In the present circumstances, it would be very useful to be able to show directly that arbitrary plurisubharmonic functions can be expressed suitably in terms of the special plurisubharmonic functions of the form loglfl for holomorphic functions f, for that would yield a direct proof of the theorem that pseudoconvexity is the same as holomorphic convexity, thereby finessing the digression through Dolbeault cohomology in section O. It is possible to express arbitrary plurisubharmonic functions in terms of these special plurisubharmonic functions by lattice rather than linear operations, at least in a domain of holomorphy. But present proofs require the equivalence of pseudoconvexity and holomorphic convexity. The interest in such a representation was indicated in the discussion in the book on several complex variables by Bochner and Martin; a proof was outlined by H. J. Bremermann, but the version to be given here follows that of N. Sibony. It may be recalled from the discussion in section L that the supremum of any collection of plurisubharmonic functions locally bounded from above is a nearly plurisubharmonic function v, the upper envelope of which is a plurisubharmonic function v*. The two functions v and v* coincide outside a set of measure zero and so represent the same locally L l-function. 9. THEOREM. If D is a domain of holomorphy in en, the set of plurisubharmonic functions of the form (sup{cj logl./jl} )*, where Cj > 0 are constants and ./j are holomorphic functions in D, is dense in the cone of plurisubharmonic functions in D in the topology of local L l-convergence. Proof.

Considering the cone of plurisubharmonic functions in the topology of local

L l-convergence implicitly excludes those functions that are identically equal to - OCJ on any connected component of D. It follows immediately from Theorem Kll that

the Coo plurisubharmonic functions are dense in this cone of plurisubharmonic functions, as a simple consequence of the monotone convergence theorem for the Lebesgue integral. Hence, it is sufficient just to show that any smooth plurisubharmonic function in D can be approximated arbitrarily well in the topology of

Q Plurisubharmonic and Holomorphic Functions

183

local U-convergence by the special plurisubharmonic functions ofthe statement of the theorem. Actually something a bit stronger will be demonstrated. Consider then a Coo plurisubharmonic function u in the domain D, and in terms of this function introduce the auxiliary open subset

D=

{(Z, w) E D xC: Iwl
O. For any point A E K, the power series 00

g(A, w) =

L

e·u(A)w·

.=0

in the complex variable w clearly has radius of convergence R(A) = e-u(A). This series can thus be viewed as representing a holomorphic function on the submanifold (A x C) n D of the domain of holomorphy D, so by Theorem E7 (in view of what was established in Theorem PI) it can be extended to a holomorphic function g(Z, w) in all of D. At any point ZED the latter function has a power series expansion 00

g(Z, w) =

L

.=0

f.(Z)w·

valid in (Z x C) n D and hence with radius of convergence R(Z) for which R(Z) ~ The coefficients f.(Z) are hoi om orphic functions in D, with the property that f.(A) = e·u(A), of course. As is quite familiar, the radius of convergence is given explicitly in terms of the coefficients of the power series by Hadamard's formula e-u(Z).

_1_) = lim sup yllf.(z)1 R(Z

Consequently,

•

184

Volume I

Function Theory

s~p ~ 10glfv(Z)1

(2)

1 -) = I·1m sup -1 Ioglfv(A)1 u(A) = log - (

(3)

u(Z)

~ log R;Z) =

lim

for all ZED, while

RA

v

v

The assertion (2) can be refined considerably with just a bit more effort. Choose an open neighborhood U of the compact set K having compact closure U c D, and a value r > 0 sufficiently small that ~(Zo,

whenever

r) s; U

Zo E K

and u(Z) ~ u(Zo)

e

+4

whenever

Zo E K

and

Z E ~(Zo; r)

It is a simple consequence of the Cauchy inequalities of Theorem AS that the functions (l/v) 10glfv(Z)1 are uniformly bounded for all v ~ 1 and Z E U. SO choose a constant M for which 1

-loglfv(Z)1 ~ M V

for

v ~ 1 and

Z EU

and u(Z) ~ M

for

Z EU

The functions

are nearly plurisubharmonic on U by Theorem L9, satisfy Igv(Z)1 ~ M for all Z E U, and from their definition are monotonically decreasing in v with limit 1 g(Z) = lim gv(Z) = lim sup -loglfv(Z)1 v v v

Hence from (2), g(Z) ~ u(Z)

The subsets

for

Z EU

Q

U. =

{Z

E

U: g.(Z) < u(Z)

+

Plurisubharmonic and Holomorphic Functions

185

i}

are then measurable subsets of U, and U. ~ U.+ 1 while U. U. = U. There is consequently an index Vo such that the Lebesgue measure IU - U.I of the set U - U. satisfies 6

IU - U.I < 8M (nr2)"

For any point Zo g:(Zo)

E

whenever

v> Vo

K the plurisubharmonic function g:(Z) satisfies

~ 1~(Zo; r)r1

r

g:(Z) dV(Z)

J.1(zo;r)

where dV(Z) is Lebesgue measure in C". This is an immediate consequence of the basic integral inequality J(2), as observed earlier in the proof of Theorem K6, for instance. Then if v > Vo and Zo E K, g.(Zo) - u(Zo) ~ g:(Zo) - u(Zo)

r

~ 1~(Zo; r)I- 1

[g:(Z) - u(Zo)] dV(Z)

J .1(Zo;r)

~ 1~(Zo; r)I~ 1~(Zo; r)I-

1

r

[g.(Z) - u(Z)

+

J .1(Zo;r)

1{

r

iJ

[g.(Z) - u(Z)

J.1(zo;r)nu,

+

Jr

[g.(Z) - u(Z)

.1(Zo;r)n(U-U,)

~ 1~(Zo; r)r1 { r

f

.1(Zo;r)n(U-u,)

iJ

~ dV(Z) +

J.1(zo;r)nu,

+

+

dV(Z)

+

iJ

dV(Z)

dV(Z)}

r

J.1(zo;r)n(u-u,)

6}

-dV(Z)

4

1 -logl!.(Z)1 v

whenever

v>

Vo

and

Z

E

K

(4)

the desired strengthening of (2). Finally from (3) and (4) it is evident that there is an index v for which u(Z)

1

+ e > -logl!.(z)1 v

for all

Z

E K

for all

Z

E UA

and

1

u(A) - e < -logl!.(A)1 v

Hence, of course, 1

u(Z) - e < -loglJ.(Z)1 V

where UA is some open neighborhood of A. The argument can be applied to each point A E K, and finitely many of the associated open sets UA will cover K. This will yield a finite collection of holomorphic functions jj in D and constants cj for which u(Z)

+ e > sup {cj 10gljj(Z)I} >

u(Z) - e

j

for all points Z E K, showing that u(Z) can be approximated arbitrarily well in the topology of local uniform convergence and hence in the topology of local L1-convergence by functions of the desired form, and thereby completing the proof. As is quite evident, the proof actually yields the stronger assertion that any Coo plurisubharmonic function in D can be approximated uniformly on compact subsets of D by special plurisubharmonic functions of the form SUPj{ cj logljjl}, in which jj are holomorphic in D and Cj ~ 0, with cj > 0 for only finitely many indices. The same is, of course, true for merely continuous plurisubharmonic functions, since they can be approximated similarly by COO plurisubharmonic functions as a consequence of Theorem K 11. The stronger result is not terribly natural, though, since the functions C 10gl!1 may well fail to be continuous in D and so do not lie in a space for which the topology of local uniform convergence is natural. The theorem itself does show that holomorphic and plurisubharmonic functions are really quite closely related, at least in domains of holomorphy; that may make it clearer why pseudoconvexity and holomorphic convexity amount to the same thing. The result of the theorem is not true for arbitrary open subsets of en, though. That is evident from the example at the end of section M of a subset D £: en

Q Plurisubharmonic and Holomorphic Functions

187

such that the envelope of holomorphy of D is properly larger than D, but there are plurisubharmonic functions in D that cannot be extended to plurisubharmonic functions in any properly larger domain. There is a considerable body of further knowledge about plurisubharmonic functions and their role in the study of holomorphic functions, but there are many interesting open questions as well.

R Pseudoconvex Sets with Smooth Boundaries

For open subsets of en with smooth boundaries, there are additional useful alternative characterizations of pseudoconvexity and various other properties closely related to pseudoconvexity. Recall that an open subset D £;; ~m is said to have a smooth boundary of class C iffor each point A E aD there are an open neighborhood UA of A in ~m and a C diffeomorphism FA: UA ~ B(O; 1) £;; ~m such that FA(Ua n D) = {X = (x l' ... , xm) E B(O; 1): Xm < O}. A C diffeomorphism is a homeomorphism FA with the property that all the component functions of both FA and the inverse mapping FAl have continuous partial derivatives of all orders ~r. The condition that an open subset D £;; ~m has a smooth boundary of class C will be indicated briefly by writing aD E c. Although this is a significant smoothness condition even for r = 0, the interest here lies more in subsets with boundaries of class C for r ~ 1indeed, usually for r ~ 2. If r ~ 1, it is clear that alternatively an open subset D £;; ~m has a smooth boundary of class C if and only if for each point A E aD there are an open neighborhood UA of A in ~m and a real-valued function PA of class C in UA such that dPA #- 0 in UA and UA n D = {X E UA : PA(X) < O}. The functions PA are called local defining functions of class C for the subset D. Indeed, the condition that dPA #- 0 in UA just means that the matrix of first partial derivatives of the function PA is nonsingular at each point of UA' and is precisely the condition that the function PA can be viewed as one of a system oflocal coordinate functions of class cr at each point of UA • Thus, it is clear that these two characterizations of sets with smooth boundaries of class C for r ~ 1 are actually equivalent. The convention that sets will be described in terms of local defining functions PA by inequalities of the form PA < 0 will be followed systematically here. The particular choice of convention is not important, although there are some minor conveniences in the choice made here, but consistency is important. Note that if PA are local defining functions for a subset D £;; ~m, then their negatives - PA are local defining functions for the complementary subset ~m - D. Note also that if P and P' are any two local defining functions of class C, r ~ 1, for an open subset D £;; ~m in a neighborhood U, then P can be viewed as a local coordinate function near any point of U and the Taylor expansion of p'

188

R Pseudoconvex Sets with Smooth Boundaries

189

in these local coordinates leads to an expression of the form p' = hp, where h is a function of class Cr - 1 and h(X) > 0 at every point X E U. If D s;; IR mis an open subset with boundary aD E C, then the boundary aD is evidently a differentiable submanifold of class C and of dimension rn - 1 contained in IRm. If r ~ 1, this submanifold aD s;; IRm has a well-defined tangent space at each point A E aD, and that tangent space can be viewed either as a linear subset of dimension rn - 1 in IRm passing through the point A or as the translate of that linear subset to the origin in IRm. In the latter case the tangent space at A is thus identified with a linear subspace 1A(aD) of dimension rn - 1 in the real vector space IRm. If p is a local defining function of class C for the subset D in an open neighborhood of A, then as is familiar 1A(aD) = { T

E

IRm :

~ tj ::j (A) = o}

(1)

This is clearly independent of the choice of local defining function. The preceding familiar differential-geometric considerations naturally also apply to open subsets D s;; en upon identifying en with 1R2n. If D s;; en is an open subset with boundary aD E C for r ~ 1, then the tangent space to the real differentiable submanifold aD at a point A E aD can be identified with a real linear subset 1A(aD) of dimension 2n - 1 in the complex vector space en. Any tangent vector TE 1A(aD) is thus viewed as a complex vector T = (tl' ... , tn ) E en, but the set of all these tangent vectors forms a real rather than a complex linear subspace of en. To rewrite the explicit expression (1) for this tangent space in complex terms, note that if p is a local defining function of class C 1 for the subset D in an open neighborhood of the point A, then for tj = uj + iVj it follows that

since p is a real-valued function. Consequently, 1A(aD) = { TE

en: Re ~ tj :~ (A) = o}

(2)

clearly emphasizing the fact that 1A(aD) is only a real linear subspace of en. Since 1A(vD) is not a complex linear subspace, i~(aD) does not coincide with ~(aD) but is another real linear subspace of dimension 2n - 1 in en. But the intersection ~(aD) n i~(aD) is clearly a complex linear subspace-indeed, is evidently a complex linear subspace of dimension n - 1 and the largest complex linear subspace of

190

Volume I

Function Theory

en lying in

1A(aD). This subspace is called the complex part of the tangent space 1A(aD) and is denoted by 1A1 .O(aD). It is clear that T E 1A1 .O(aD) precisely when T E 1A(aD) and iT E 1A(aD). It thus follows immediately from (2) that

(3) clearly showing that 1A1 .O(aD) is a complex linear subspace of en. If D s;; en is an open subset with boundary aD E Cr for r ~ 2, then it is possible to consider in addition to the linear approximation to aD, the tangent space to the submanifold aD s;; en, also the second-order approximation to aD, involving local curvature properties of the submanifold; this too is of considerable interest from the point of view of complex analysis. If p is a local defining function of class cr, r ~ 2, for the subset D in an open neighborhood of a point A E aD, the natural quadratic expression to consider is the Levi form Lp(A; T); however, this form depends on the choice of the local defining function p in a nontrivial way. If p' is another local defining function of class cr, r ~ 2, for the subset D in an open neighborhood of the point A, then as already noted p' = hp, where h is a function of class Cr - 1 and h(Z) > O. Even though the function h may only be of class C 1 , nonetheless since p(A) = p'(A) = 0 clearly

Consequently, Lp'(A; T) = h(A)Lp(A; T)

~ ap _ ah + 2 Re L... tr~-(A) tk = jk

uZj

(A)

(4)

uZk

which is a rather nontrivial relation. However, if the vector T is restricted to lie in the complex tangent space 1A1 .O(aD) defined by (3), then Lp'(A; T) = h(A)Lp(A; T), where h(A) > O. Thus, the restriction of the Levi form Lp(A; T) to vectors T E 1A1.0(aD) is a Hermitian form on the complex vector space 1A1 .O(aD) and is uniquely determined up to a positive scalar factor by the boundary aD alone, independent of the choice of a local defining function for the set D. This class of Hermitian forms on 1A1.O(aD), all of which differ by only a positive scalar factor, will be called the Levi form of the boundary aD at the point A. 1. DEFINITION. An open subset D s;; en with boundary aD E c 2 is pseudoconvex in the sense of Levi if the Levi form of the boundary aD is positive semidefinite at each point A E aD. D is strictly pseudoconvex in the sense of Levi if the Levi form of the boundary aD is actually positive definite at each point A E aD. It is clear that these notions are really well defined, since multiplication by a positive constant preserves the classes of positive semidefinite and positive definite

R Pseudoconvex Sets with Smooth Boundaries

191

Hermitian forms. A strictly Levi pseudo convex set is sometimes also referred to as a strongly Levi pseudoconvex set. As might be guessed from the terminology, this notion of pseudoconvexity is equivalent to the notion discussed in earlier sections, for subsets with sufficiently smooth boundaries, of course. A convenient way to demonstrate this equivalence involves using the distance function dD , which played a prominent role in the earlier discussion of pseudoconvexity, as a local defining function for the set D. If should be observed, however, that the function dD as introduced in Definition G3 and used so far has been defined only in the set D itself. But it is a simple matter to extend this function to one defined throughout e" by setting if Z E aD if z¢i5

(5)

e", and moreover that - dD is the continuous analogue of a local defining function for the subset D near each point of the boundary aD. It only fails to be a local defining function by not having the property that dPA =F 0 near aD, a property that of course it cannot have ifthe boundary is nondifferentiable. However, the extended function -d D is a proper local defining function whenever the boundary of D is sufficiently smooth. It is obvious that the function dD as so extended is continuous in all of

If D ~ e" is an open subset with boundary aD E C 2 , then the extended function - dD is a local defining function of class c 2 for the subset D near each boundary point.

2. LEMMA.

Proof. The complex structure in e" plays no role here, and it is even convenient to ignore it altogether; hence, e" will be identified with ~211, with real coordinates X = (Xl' ... , X 211 ) or Y = (YI' ••• , Y211)' In order to prove the desired result, it is sufficient merely to show that the extended function dD is of class C 2 near aD and has a nontrivial differential at each point of aD. For any point A E aD, select some local defining function P of class C 2 for the subset D in an open neighborhood U of the point A. It is familiar from elementary differential geometry that the gradient vector Vp(X) = (ap(X)/ax I , ... , ap(X)/ax 211 ) is normal to the submanifold aD at each point X E U n aD. Indeed, in view of the convention for local defining functions adopted here, Vp(X) is a normal vector directed outward from D. It is also familiar that whenever X E Un aD and Y = X + t· Vp(X) where t E ~ and It I is sufficiently small, then the ball B(Y; r) passing through X is tangent to the submanifold aD at X, and that ball is either entirely contained within D or entirely exterior to D depending on the sign of t. Thus, dD(y) = ±r = ± II Y - XII. Now the mapping F: (U n aD) x ~ -+ ~211 defined by F(X, t) = X + t· Vp(X) clearly has a nonsingular Jacobian matrix whenever t = 0, so if the neighborhood U of A is suitably chosen, then F establishes a C 1 diffeomorphism F: (U n aD) x ( - e, e) -+ U for some sufficiently smalle. Any point Y E U can therefore be written in a unique way in the form Y = F(X, t) = X + t· Vp(X), where X E Un aD and Itl < e. The mapping P: U -+ Un all defined by P(Y) = X is then a C l projection mapping, and if U and e are sufficiently small, dD(Y) = ± II Y - P(Y)II whenever Y E U, for an appropriate

192

Volume I

Function Theory

choice of sign depending on whether Y E D or Y ~ D. It is clear from this that dD is a function of class C 1 in U except perhaps at Un aD, and indeed that when YE U - UnaD,

(6) On the other hand, since P(Y) E aD, necessarily p(P(Y» = 0 for all points Y and upon differentiating it follows that

o=

a

L

-;- p(P( Y» = uYi k

where X=P(Y). But if consequently,

ap(X) a -:1- -;-

uXk

uYi

y~aD,

E

U,

Pk( Y)

then Y-P(Y)=t·Vp(X), where t#O, and

Substituting this into (6) shows that

But that just means that in U - Un aD, the gradient vector of the function dD(Y) is the unit vector parallel to Y - P(Y) = tV p(P(Y» and pointing toward the interior of D and hence that Vp(P(Y» VdD(Y) = -IIVp(P(Y»1I

The vector field VdD(Y) is therefore of class C 1 and is nowhere zero, and that suffices to conclude the proof of the lemma. Another useful auxiliary observation here is the following, which is really just a straightforward calculation. 3. LEMMA. If P is a real-valued function of class C 2 in an open neighborhood of the origin o in en, then as Z approaches 0,

R Pseudoconvex Sets with Smooth Boundaries

193

where P2P(0; Z) is the complex polynomial of degree 2 in the variables Z defined by

and Lp(O; Z) is the Levi form of p at the origin.

Since p is of class C 2 near the origin, it follows from the usual Taylor expansion of the function r at the origin in terms of the real coordinates Xj' Yj in IC" = ~2" that as Z approaches 0, Proof.

p(Z) = p(O)

+ L [aa p (0)· Xj + aa p (0)· Yj] j

Xj

Yj

+ 7)/2 and Yj = (Zj - zj)/2i, then grouping together those terms in the preceding formula involving the same variables Zj and Zk and recalling the definitions of the operators a/azj and a/azk show that the preceding formula can be rewritten as

If Xj = (Zj

p(Z) = p(O)

ap (O)·Zj + aap (O)·Zj ] + L [-a j

Zj

Zj

since p is real-valued, and that suffices for the proof. Applying the two preliminary results leads to the following basic observation.

194

Volume I

Function Theory

4. THEOREM. An open subset D ~ en with boundary aD E it is pseudoconvex in the sense of Levi.

c 2 is pseudoconvex if and only if

Proof. First suppose that D is pseudoconvex, so that by Theorem M3 the function -log d is plurisubharmonic in D, where to simplify the notation, d stands for dD • The function d is of class C 2 in D near the boundary aD-indeed, naturally extends to a function of class C 2 in a full open neighborhood of the boundary aD in en by Lemma 2. So by Theorem K8 the condition that -log d be plurisubharmonic is that its Levi form be positive semidefinite-that is, that

o ~ L( -log d)(Z; A) ~

a2 log d(Z)

jk

aZj aZ k

~ - L.

-

_

ajak

for all points ZED sufficiently near the boundary aD and all vectors A E en. In particular, therefore (7)

whenever ZED is sufficiently near the boundary and the vector A satisfies O. It follows from continuity that (7) holds in the limit for all points Z E aD whenever A E Tz1.0(aD); but that isjust the condition that D be pseudoconvex in the sense of Levi, since - d is a local defining function of class C 2 for the subset D near each boundary point of Lemma 2. Next suppose that D is not pseudoconvex. Since the pseudoconvexity of D is a local property of aD in view of Theorem M8, it follows from Theorem M3 that there must be points in D arbitrarily near the boundary at which the function -log d is not plurisubharmonic, where again d is used in place of d D and denotes the extension of dD to a function defined throughout en. Choose such a point A E D at which d is of class C 2 , as is possible in view of Lemma 2. Since d is not plurisubharmonic at A, it follows from Theorem K8 that

L aj aD(Z)/azj =

d Ljk aa 10g a- (A)cjck = Zk 2

L( -log d)(A; C) = -

-2r < 0

Zj

for some vector C E en and some positive real number r, or equivalently that 0 2 log d(A

at at

+ tC) I = 2r > 0 1=0

It follows from Lemma 3 that the Taylor expansion of the real-valued function log d(A + tC) of the point tEe near the origin can be written in the form

R Pseudoconvex Sets with Smooth Boundaries

log d(A

+ tC) =

log d(A)

for some complex constants sufficiently small 8 > 0, then log d(A

0(

195

+ Re[O(t + pt 2] + 2rltl2 + 0(ltI2) and

p. Consequently, whenever It I < 8 for some

+ tC) ~ log d(A) + Re[O(t + pt 2] + rltl 2

or equivalently (8)

Choose a boundary point BEaD nearest A, so that d(A) = liB - A II, and introduce the holomorphic mapping F: C -+ en defined by F(t) = A

+ tC + (B -

A)e at+ Pt2

Note that F(O) = BEaD, but that whenever 0 < It I < d(A

+ tC) >

8,

then erltl2 > 0, so that by (8),

liB - Allleat+Pt21

and consequently F(t) E D. The composite function d(F(t)) is thus nonnegative for It I < 8; indeed, from the triangle inequality and (8) it follows that whenever It I < 8, then d(F(t)) ~ d(A

+ tC) -

II(B - A)eat+Pt211

The function d(F(t)), which is of class C 2 near t = 0 since F(O) = BEaD and by Lemma 2 the extended function is of class C 2 near aD, thus attains a local minimum at t = 0 but is not 0(ltI2). Therefore

a

at d(F(t))lt=o = 0

a2

at at d(F(t))lt=o > 0

Writing F = (fI' ... , In) and using the complex form of the chain rule show, since !j are holomorphic, that

I

j

iJd -(B)· !j'(0) = 0 iJzj

Since - d is a local defining function for D, this shows that D is not pseudoconvex in the sense of Levi, and thereby concludes the proof. The notion of pseudoconvexity given in Definition 1 was introduced by E. E. Levi (in Annali di Mat. pura ed appl., vol. 17, 1910) and antedates the more general

196

Volume I

Function Theory

notion given in Definition MI. Levi showed that every domain in C n that is pseudoconvex in his sense is at least locally holomorphically convex, thus raising the question whether a locally holomorphically convex domain is actually holomorphically convex-that is to say, whether holomorphic convexity is actually a local property. This problem has subsequently commonly been known as the Levi problem. The solution of the Levi problem discussed earlier here involved the more general notion of pseudoconvexity and so was valid for arbitrary open subsets of Cn (or of Riemann domains) without assuming any boundary regularity. The consideration of domains having smooth boundaries leads to a vast array offurther fascinating questions, such as the boundary behavior of various classes of hoI om orphic functions (including generalizations of the classical HP spaces in the unit disc in one variable) or of holomorphic mappings between smoothly bounded domains, and the possibilities of more general Cauchy integral formulas involving integration on the full boundaries. Even to begin to discuss these questions, which have been actively investigated with a wide variety of major results scattered throughout the literature, would extend the present volume excessively. The reader must be referred to several other treatises dealing with these matters.

Bibliography

1. Abhyanker, S. S. Local Analytic Geometry. Academic Press; New York, 1964. 2. Andreotti, A., and Stoll, W. Analytic and Algebraic Dependence of Meromorphic Functions. Lecture Notes in Math., 234. Springer-Verlag; Heidelberg, 1970. 3. Baily, W. L. Several Complex Variables. University of Chicago Press; Chicago, 1957. 4. Banica, c., and Stanasila, O. Algebraic Methods in the Global Theory of Complex Spaces. J. Wiley; London, 1976. 5. Beals, R., and Greiner, P. Calculus on Heisenberg Manifolds. Annals of Math. Studies, 119. Princeton University Press; Princeton, NJ, 1988. 6. Behnke, H., and Thullen, P. Theorie der Funktionen mehrerer komplexer Veriinderlichen. (2 Aufl.) Ergebnisse Math., 51. Springer-Verlag; Heidelberg, 1970. 7. Bergman, S. The Kernel Function and Conformal Mapping. Math. Surveys American Math. Society; Providence, RI, 1950. 8. Bergman, S. Sur les Fonctions Orthogonales de Plusieurs Variables Complexes avec Applications Ii la Theorie des Fonctions Analytiques. Interscience; New York, 1941. 9. Bers, L. Introduction to Several Complex Variables. New York University Press; New York, 1964. 10. Bochner, S., and Martin, W. T. Several Complex Variables. Princeton University Press; Princeton, NJ, 1948. 11. Cartan, H. Seminaire Henri Cartan. 1951/52: Fonctions Analytiques de plusieurs variables complexes. 1953/54: Fonctions automorphes et espaces analytiques. 1960/61: Familles d'espaces complexes et fondements de la geometrie analytique. W. A. Benjamin; New York, 1967. 12. Cazacu, C. A. Theorie der Funktionen mehrerer komplexer Veriinderlichen. VEB Deutscher Vorlag der Wissenschaften; Berlin, 1975. 13. Chern, S. S. Complex Manifolds Without Potential Theory. Van Nostrand; Princeton, NJ,1962. 14. Coleff, N. R., and Herrera, M. E. Les courants residuel associes liune forme meromorphe. Lecture Notes in Math., 633. Springer-Verlag; Heidelberg, 1978. 15. Ehrenpreis, L. Fourier Analysis in Several Complex Variables. Wiley-Interscience; New York, 1970. 16. Fischer, G. Complex Analytic Geometry. Lecture Notes in Math., 538. Springer-Verlag; Heidelberg, 1976. 17. Folland, G., and Kohn, J. J. The Neumann Problem for the Cauchy-Riemann Complex. Annals of Math. Studies, 75. Princeton University Press; Princeton, NJ, 1972. 197

19B

Bibliography

18. Fornaess, J. E. Recent Developments in Several Complex Variables. Annals of Math. Studies, 100. Princeton University Press; Princeton, NJ, 1981. 19. Fornaess, 1. E., and Stens0nes, B. Lectures on Counterexamples in Several Complex Variables. Math. Notes, 33. Princeton University Press; Princeton, NJ, 1987. 20. Fuks, B. A. Introduction to the Theory of Analytic Functions of Several Complex Variables. American Math. Society; Providence, RI, 1963. 21. Gamelin, T. W. Uniform Algebras. Prentice-Hall; Englewood Cliffs, NJ, 1969. 22. Grauert, H., and Frtizsche, K. Several Complex Variables. Graduate Texts in Math., 38. Springer-Verlag; New York, 1976. 23. Grauert, H., and Remmert, R. Analytische Stellenalgebren. Grundlehren Math. Wissenschaften, 176. Springer-Verlag; Heidelberg, 1971. 24. Grauert, H., and Remmert, R. Theorie der Steinschen Riiumen. Grundlehren Math. Wissenschaften, 236. Springer-Verlag; Heidelberg, 1979. 25. Greiner, P., and Stein, E. M. Estimates for the a-Neumann problem. Math. Notes, 19. Princeton University Press; Princeton, NJ, 1977. 26. Griffiths, P. A. Entire Holomorphic Mappings in One and Several Variables. Annals of Math. Studies, 85. Princeton University Press; Princeton, NJ, 1976. 27. Gunning, R. C. Lectures on Complex Analytic Varieties: The Local Parametrization Theorem. Mathematical Notes, 10. Princeton University Press; Princeton, NJ, 1970. 28. Gunning, R. C. Lectures on Complex Analytic Varieties: Finite Analytic Mappings. Math. Notes, 14. Princeton University Press; Princeton, NJ, 1974. 29. Gunning, R. C., and Rossi, H. Analytic Functions of Several Complex Variables. PrenticeHall; Englewood Cliffs, NJ, 1965. 30. Henkin, G. M., and Leiterer, J. Theory of Functions on Complex Manifolds. Birkhiiuser; Boston, 1984. 31. Herve, M. Several Complex Variables, Local Theory. Oxford University Press; Oxford, 1963. 32. Hormander, L. An Introduction to Complex Analysis in Several Variables. Van Nostrand; Princeton, NJ, 1966. 33. Hua, L. K. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Math. Monographs, 6. American Mathematical Society; Providence, RI, 1963. 34. Kaup, L., and Kaup, B. Holomorphic Functions of Several Variables. Walter de Gruyter; Berlin, 1983. 35. Kobayashi, S. Hyperbolic Manifolds and Holomorphic Mappings. Marcel Dekker; New York,1970. 36. Krantz, Steven G. Function Theory of Several Complex Variables. J. Wiley; New York, 1982. 37. Laufer, H. B. Normal Two-Dimensional Singularities. Annals of Math. Studies, 71. Princeton University Press; Princeton, NJ, 1971. 38. Lelong; P. Fonctionelles analytiques et fonctions entieres (n variables). Seminaire de Math. superieure, 28. Les Presses de L'Universite de Montreal; Montreal, 1968. 39. Lelong, P. Fonctions Plurisousharmoniques et Formes Differentielles Positives. Gordon and Breach; Paris, 1968. 40. Malgrange, B. Lectures on the Theory of Functions of Several Variables. Tata Institute; Bombay, 1958. 41. Milnor, J. Singular Points on Complex Hypersurfaces. Annals of Math. Studies, 61. Princeton University Press; Princeton, NJ, 1968. 42. Narasimhan, R. Analysis on Real and Complex Manifolds. Masson; Paris, 1973. 43. Narasimhan, R. Introduction to the Theory of Analytic Spaces. Lecture Notes in Math., 25. Springer-Verlag; Heidelberg, 1966.

Bibliography

199

44. Narasimhan, R. Several Complex Variables. University of Chicago Press; Chicago, 1971. 45. Oka, K. Sur les Fonctions Analytiques de Plusieurs Variables. Iwanami Shoten; Tokyo, 1961. 46. Osgood, W. F. Lehrbuch der Funktionentheorie, Bd II. Chelsea; New York, 1965. 47. Pflug, R. P. Holomorphiegebiete, pseudoconvexe Gebiete, und das Levi-Problem. Lecture Notes in Math., 432. Springer-Verlag; Heidelberg, 1975. 48. Range, R. M. Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics, 108. Springer-Verlag; New York, 1986. 49. Rothstein, W. Vorlesungen uber Einfiihrung in die Funktionentheorie mehrerer komplexer Veriinderlichen, Bd. I, II. Aschendorffsche Verlagsbuchhandlung; Munster, 1965. 50. Rudin, W. Function Theory in the Unit Ball of Grundlehren der Math. Wissenschaften, 241. Springer-Verlag; Berlin, 1980. 51. Rudin, W. Function Theory in Polydiscs. W. A. Benjamin; New York, 1969. 52. Rudin, W. Lectures on the Edge-of-the-Wedge Theorem. CBMS 6. American Mathematical Society; Providence, RI, 1970. 53. Siegel, C. L. Analytic Functions of Several Complex Variables. Institute for Advanced Study; Princeton, NJ, 1948.

en.

54. Siu, Y-T., and Trautmann, G. Gap-Sheaves and Extension of Coherent Analytic Sheaves. Lecture Notes in Math., 172. Springer-Verlag; Heidelberg, 1971. 55. Stein, E. M. Boundary Behavior of Holomorphic Functions of Several Complex Variables. Math. Notes, 11. Princeton University Press; Princeton, NJ, 1972. 56. Vladimirov, V. S. Les Fonctions de Plusieurs Variables Complexes (et leur application Ii la theorie quantique des champs). Dunod; Paris, 1967. 57. Vladimirov, V. S. Methods of the Theory of Functions of Many Complex Variables. M.I.T. Press; Cambridge, MA, 1966. 58. Wells, R. 0., Jr. Differential Analysis on Complex Manifolds. Prentice-Hall; Englewood Cliffs, NJ, 1973. 59. Wermer, M. Banach Algebras and Several Complex Variables. Graduate Texts in Math., 35. Springer-Verlag; New York, 1976. 60. Whitney, H. Complex Analytic Varieties. Addison-Wesley; Reading, MA, 1972.

About this book: Introduction to Holomorphlc Functions of Several Variables, Volumes I-III provide an extensive introduction to the Oka-Cartan theory of holomorphic functions of several variables and holomorphic varieties . Each volume covers a different aspect and can be read independently. . • Volume I: Function Theory (ISBN: 0-534-13308-8) • Volume II: Local Theory (ISBN : 0-534-13309-6) • Volume III: Homological Theory (ISBN: 0-534-13310-X)

Other titles from the Wadsworth & Brooks/Cole Mathematics Series Complex Variables, Second Edition Stephen D. Fisher This undergraduate text for majors in engineering and mathematics offers a direct route to the most important topics in the theory and applications of complex variables. Thoroughly updated and revised . Contents: 1. The complex plane . 2. Basic properties of analytic functions . 3. Analytic functions as mappings . 4. Analytic and harmonic functions in applications . 5. Transform methods. 1990. Clothbound. 448 pages . ISBN : 0-534-13260-X.:

ISBN 0-534-13308-8 90000

9 780534 133085